Graduate Descenthttps://timvieira.github.io/blog/2021-03-25T00:00:00-04:00Fast rank-one updates to matrix inverse?2021-03-25T00:00:00-04:002021-03-25T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2021-03-25:/blog/post/2021/03/25/fast-rank-one-updates-to-matrix-inverse/<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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<p>Let $A \in \mathbb{R}^{n \times n}$. The <a href="https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula">Sherman–Morrison formula</a> suggests a computational shortcut to update a matrix inverse subject to a rank-one update, i.e., an additive change of the form $A + u \, v^\top$ where $u, v \in \mathbb{R}^n$:</p>
$$
(A + u \, v^\top)^{-1}
$$<p>Suppose we have precomputed $B = A^{-1}$, the shortcut is
$$
(A + u \, v^\top)^{-1} = B - \frac{1}{1 + v^\top B\, u} B\, u\, v^\top B
$$</p>
<p>Implemented carefully, it runs in $\mathcal{O}(n^2)$, which is better than $\mathcal{O}(n^3)$ from scratch. Ok, <em>technically</em> matrix inversion can run faster than cubic, but you get my point.</p>
<p>Below is a direct translation of equations into numpy:</p>
<div class="highlight"><pre><span></span><span class="n">B</span> <span class="o">-=</span> <span class="n">B</span> <span class="o">@</span> <span class="n">u</span> <span class="o">@</span> <span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">B</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">B</span> <span class="o">@</span> <span class="n">u</span><span class="p">)</span>
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<p><em>However</em>, it does <strong>NOT</strong> run in $\mathcal{O}(n^2)$. Do you see why?</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span><span class="o">,</span> <span class="nn">scipy</span>
<span class="kn">from</span> <span class="nn">numpy.linalg</span> <span class="kn">import</span> <span class="n">inv</span>
<span class="k">def</span> <span class="nf">slowest</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">c</span><span class="p">):</span>
<span class="k">return</span> <span class="n">inv</span><span class="p">(</span><span class="n">A</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">outer</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">slow</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">):</span>
<span class="k">return</span> <span class="n">B</span> <span class="o">-</span> <span class="n">B</span> <span class="o">@</span> <span class="n">u</span> <span class="o">@</span> <span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">B</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">B</span> <span class="o">@</span> <span class="n">u</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">fast</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">):</span>
<span class="k">return</span> <span class="n">B</span> <span class="o">-</span> <span class="p">(</span><span class="n">B</span> <span class="o">@</span> <span class="n">u</span><span class="p">)</span> <span class="o">@</span> <span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">B</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">B</span> <span class="o">@</span> <span class="n">u</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal</span> <span class="kn">import</span> <span class="n">iterview</span><span class="p">,</span> <span class="n">timers</span>
<span class="k">def</span> <span class="nf">workload</span><span class="p">():</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">iterview</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">3.75</span><span class="p">,</span> <span class="mi">10</span><span class="p">)):</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">)</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">inv</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="k">yield</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="n">timers</span><span class="p">()</span>
<span class="k">for</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="ow">in</span> <span class="n">workload</span><span class="p">():</span>
<span class="k">with</span> <span class="n">T</span><span class="p">[</span><span class="s1">'slowest'</span><span class="p">](</span><span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">slowest</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)</span>
<span class="k">with</span> <span class="n">T</span><span class="p">[</span><span class="s1">'slow'</span><span class="p">](</span><span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">slow</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)</span>
<span class="k">with</span> <span class="n">T</span><span class="p">[</span><span class="s1">'fast'</span><span class="p">](</span><span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">):</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">fast</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)</span>
<span class="c1"># pro tip: always check correctness when optimizing your code!</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
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<pre>100.0% (10/10) [======================================================] 00:00:24
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">T</span><span class="o">.</span><span class="n">plot_feature</span><span class="p">(</span><span class="s1">'n'</span><span class="p">);</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">T</span><span class="o">.</span><span class="n">compare</span><span class="p">()</span>
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<pre>fast is 10.8188x faster than slow (<span class="ansi-yellow-fg">p=0.08099</span>, mean: slow: 0.552587, fast: 0.0510765)
fast is 15.0779x faster than slowest (<span class="ansi-green-fg">p=0.03201</span>, mean: slowest: 0.770126, fast: 0.0510765)
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<p><strong>What's going on?</strong></p>
<p>As you can see in the plots, the <code>slow</code> and <code>slowest</code> algorithms run at similar speeds despite all that extra effort to implement the fancy formula. As expected, when carefully implemented, the rank-one update method is lightning fast.</p>
<p>I suspect that this mistake is quite common. By default, multiplication is a left-associative operation. This case results in an unnecessary matrix-matrix product instead of an outer-product of two matrix-vector products.</p>
<p>Looking forward, I suggest that authors put explicit parenthesis in the rank-one update formula:</p>
$$
(A + u \, v^\top)^{-1} = B - \frac{1}{1 + v^\top B\, u} \left(B\, u\right)\, \left(v^\top B\right)
$$<p>So, coders, please be careful when translating math into code. Mathers, please help us coders out by putting parens in equations.</p>
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<h3 id="Bonus-🔥">Bonus 🔥<a class="anchor-link" href="#Bonus-🔥">¶</a></h3>
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<p>To make this method really fast, we can use specialized <a href="https://en.wikipedia.org/wiki/Basic_Linear_Algebra_Subprograms">BLAS</a> routines for rank-one updates. Thanks to
Dr. Nikos Karampatziakis for suggesting it on <a href="https://twitter.com/eigenikos/status/1375143742064062464">Twitter</a>.</p>
<p>Specifically, <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.blas.dger.html">DGER</a>, which computes
$$
B \gets B + \alpha \, x\, y^\top
$$</p>
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<p>The following methods have a slightly different API as they do updates to $B$ <em>in place</em>.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">fastest</span><span class="p">(</span><span class="n">B</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">):</span>
<span class="c1"># Warning: `overwrite_a=True` silently fails when B is not an order=F array!</span>
<span class="k">assert</span> <span class="n">B</span><span class="o">.</span><span class="n">flags</span><span class="p">[</span><span class="s1">'F_CONTIGUOUS'</span><span class="p">]</span>
<span class="n">Bu</span> <span class="o">=</span> <span class="n">B</span> <span class="o">@</span> <span class="n">u</span>
<span class="n">alpha</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">Bu</span><span class="p">)</span>
<span class="n">scipy</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">blas</span><span class="o">.</span><span class="n">dger</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">Bu</span><span class="p">,</span> <span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">B</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="n">B</span><span class="p">,</span> <span class="n">overwrite_a</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">faster</span><span class="p">(</span><span class="n">B</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">):</span>
<span class="n">Bu</span> <span class="o">=</span> <span class="n">B</span> <span class="o">@</span> <span class="n">u</span>
<span class="n">np</span><span class="o">.</span><span class="n">subtract</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">Bu</span> <span class="o">@</span> <span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">B</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">v</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">Bu</span><span class="p">),</span> <span class="n">out</span><span class="o">=</span><span class="n">B</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">T2</span> <span class="o">=</span> <span class="n">timers</span><span class="p">()</span>
<span class="k">for</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="ow">in</span> <span class="n">workload</span><span class="p">():</span>
<span class="n">ref</span> <span class="o">=</span> <span class="n">fast</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="k">with</span> <span class="n">T2</span><span class="p">[</span><span class="s1">'faster'</span><span class="p">](</span><span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">):</span>
<span class="n">faster</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">ref</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span> <span class="n">order</span><span class="o">=</span><span class="s1">'F'</span><span class="p">,</span> <span class="n">copy</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">with</span> <span class="n">T2</span><span class="p">[</span><span class="s1">'fastest'</span><span class="p">](</span><span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">):</span>
<span class="n">fastest</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">ref</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
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<pre>100.0% (10/10) [======================================================] 00:00:12
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">T2</span><span class="o">.</span><span class="n">plot_feature</span><span class="p">(</span><span class="s1">'n'</span><span class="p">);</span>
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YxZRE6OR3hiZzcnRsI0+H3cu2WFJYd5lO0opn/n9jPcBCwBfgP44nn2eQVYLSIrRcQH3Ak8nc0vExG/iFRPfXZ/754sYzXGFIHu0RBPun0OLTXlfGKL1Y+eb9k+BzHVXHQL8Leq+pqcZ5ITVU2IyGeAHwIe4FFV3Ssin3LXPywircAOoAZIicjvAWuAJmC7+yu8wD+q6g9md2jGmIXq6NAk33u9l54xKxGaT9kmiJ0i8hywEnjIvbo/6xDXKar6LPDsjGUPp30+idP0NNMEcGWWsRljisihgQDPvjFdIvSuazrxl9szvfmQ7b/6bwIbgCOqGhKRRpxmJmOMmTP7+pwSoSfHI1zaWsOdViI0r873HMRVMxatsulzjTG58Hr3GD/eN0D/RIS17bXccXUHZaVWIjSfzpea/5f7Xg5cDbyO0x+xHvgF8K7chWaMWSx2Hhvhp/utRGihOed/AVXdqqpbgWPA1arapapXAxuBQ/MRoDGmeKkqLx8e5iducuha0cAdV3dYcigQ2TbuXaaqb0x9UdU9IrIhNyEZYxaDmSVCr13VyAevtBKhhSTbBLFPRL4JPIbzNPTHgX05i8oYU9SsROjCkG2C+A3gd4DPud9fAr6ek4iMMUXNSoQuHFklCFWNAP/bfRljzAWZWSL0vZc28741ViK0UGVbD+J64D8Dy9P3UdVVuQnLGFNs0kuETsaS3LimhfdcYiVCC1m2TUx/A/xHYCeQzF04xphioqqMheKMhKK8dGCIk+MRYokUt65vY5NVgSt42SaIcVX9fk4jMcYUFVXlQH+QvvEw/3pwiJMTEUo9Jdx77XI2LK/Pd3gmC9kmiBdF5EvAU0B0aqGq7spJVMaYBW8sFKdnNMy/HBpkYCICwOaV9SxvskI/C0W2CWKqElxX2jIFbpjbcIwxxaJvPMxPDgwwGIjiLRF+5ZIltNdXMhlLUO/35Ts8k4VsRzFtzXUgxpjiMRiI8vyb/fRPRKgo9bD10maaa8oB8Pts8r2FIttRTLXAnwLvcRf9FPgzVR3PVWDGmIWpdyzMP+/uYSgYZUlVGVsuaqTB7xT6afD7qKu0ug4LRbap/FGcim4fc79/Avhb4MO5CMoYszAdG57ku6/10jsWob2ugjs3d1BCCZOxBH6fl7rKUntaegHJNkFcpKofSfv+X0Rkdw7iMcYsUAf7A3x/j1PoZ+WSKu7etIyqcuduwfocFqZsp0wMi8ipqb3dB+fCuQnJGLPQ7OkZ55nX++gbC3Npaw2fuLbzVHIwC1e2dxC/A3zL7YsAGAV+PScRGWMWlB1HR3jpgDNd9/qOOj5ydTs+rxX6KQbZjmLaDVwpIjXu94lcBmWMKXyqyr8dGuYXbw8z5NZy+NDGdjw2XXfRyKqJSUT+u4jUqeqEqk6ISL2I/LdcB2eMKUyplPLCvgF+fmSI4WCM6y5q4lctORSdbPsgtqnq2NQXVR0FbslJRMaYgpZMKd/fc5JXj48y5k7XfeuVbVbopwhl2wfhEZEyVY0CiEgFUJa7sIwxhSiWSPHMG70c6A8yGUlww2XN3HC5TdddrLJNEI8BL4jI3+JMsfHvgG/lLCpjTMGJxJN8Z3cPbw9OEk2k2LaulS0XNeU7LJND2XZS/08ReR14HyDAf1XVH+Y0MmNMwQhGE2x/tYfjw5OkFD60sZ0rl9XlOyyTY7OZFGUfkFDVH4lIpYhUq2ogV4EZYwrDeCjOU6920z0axlsi3HF1B5e2Vuc7LDMPsh3F9FvAE8A33EXtwD/nKCZjTIEYCkb5PztOcHw4RLnXw92bOi05LCLZjmL6D8D1wASAqh4EmnMVlDEm//rGw/zTjm5OjIaoqSjlE1uWWy2HRSbbJqaoqsamJtkSES9OZ7UxpggdHw7x3dd76R0L01ZbwV2bl9Hot4GLi022CeKnIvJHQIWI3Ah8Gvhu7sIyxuTLoYEAz77hTrrX5OfuzTav0mKVbRPTg8Ag8Abw28CzwH/KVVDGmPzY0zPO915zJt27pLWaT2xZbslhEct2mGsK+Gvgr0WkAehQVWtiMqaI7Dw2wk/3O5Pureuo4w6bdG/Ry7ai3E+A29ztdwODIvJTVf393IVmjJkPqsrPDg/z8yPOpHtXL2/gV6+yeZVM9k1Mte4Mrh8G/lZVr8Z5aM4Ys4ClUsqP3xrg5cPDDAdjbLmoiQ9bcjCubBOEV0TacEqOfi+H8Rhj5kkypfxg70l2HR9lLBTj3aub+KBNumfSZJsg/gz4IXBIVV8RkVXAwfPtJCLvF5H9InJIRB7MsP4yEXlZRKIi8vnZ7GuMuXDxZIrvvtbLGz3jBMIJtl7WzLZ1bVYv2pwm207qfwL+Ke37EeAjZ98DRMQDfBW4EegGXhGRp1X1zbTNRoDPAh+6gH2NMRcgEk/y9O5ejgwGidike+YcznkHISL/yR21dLb1N4jIrWdZvQnnjuOIqsaAbwO3p2+gqgOq+goQn+2+xpjZm4wmeGJnN4cGAsRTyoc2LLXkYM7qfHcQbwDfFZEIsAvnWYhyYDWwAfgR8N/Psm87cCLtezewOcu4st5XRO4D7gPo7OzM8scbs/iMh+M8taubntEwJSJ8tKudy1pr8h2WKWDnTBCq+h3gOyKyGmcupjac+ZgeA+5T1fA5ds/UmJntsxNZ76uqjwCPAHR1ddmzGcZkMByM8tSuHnrHwlT4PPxa1zKbV8mcV7Z9EAfJolN6hm5gWdr3DqB3HvY1xqQ5OR5h+6s99I2Hqa/wcdfmTlpry/MdllkAsh3FdCFeAVaLyEoR8QF3Ak/Pw77GGNeJkRBP7uqmezTEkqoyPnHdcksOJmuzKRg0K6qaEJHP4AyP9QCPqupeEfmUu/5hEWkFdgA1QEpEfg9Yo6oTmfbNVazGFKNDA0GefcOZV2lFk5+7NnVSU2HzKpnsSTFNqdTV1aU7duzIdxjG5N3e3nGe39tP/0SES1qrufOaZVT4cnY9aBYwEdmpql2Z1mVbUe4SEXlBRPa439eLiM3makwB2nlslB/uOUn/RIR17bXcvanTkoO5INn2Qfw18BDu8wqq+jpOv4AxpkCoKj87NMRP9g8wGIhy1fJ6PnbNMspKbUZWc2GyvayoVNVfzngMP5GDeIwxF0BVeXH/ALuOjTEyGWPLRY3cun6pzatk3pFsE8SQiFyE+yyCiNwB9OUsKmNM1pIp5bm9J9nTO854KM67Vzfx/rWtNq+SeceyTRD/AedhtMtEpAd4G/h4zqIyxmQlnkzx7Bt97D8ZIBBJ8CuXNnPjmpZ8h2WKRLYPyh0B3icifqBEVQO5DcsYcz6ReJKnX3Mn3Yun2La2lesutnmVzNzJtqJcHfBJYAVObQgAVPWzuQrMGHN2k9EE21/t4fjwJMkU3HZlG1ctP+u8msZckGybmJ4Ffo4zeV8qd+EYY85nPBxn+65uut1J9+7oWsrlbbX5DssUoWwTRLnVnzYm/4aDUba/2kPPWJiKUg8f61rGCpt0z+RItgni/xOR38IpNxqdWqiqIzmJyhhzhv4JZ9K93jFn0r1f29TB0rrKfIdlili2CSIGfAn4Y6an3VZgVS6CMsac7sRIiKdf66VvLExLTTl3be6kqaos32GZIpdtgvh94GJVHcplMMaYMx0aCPL9N/roHQuzvNHP3Ztt0j0zP7JNEHuBUC4DMcac6c3eCZ5/8yQnxyOsbqnmrk026Z6ZP9n+pSWB3SLyIqf3QdgwV2NyZNfxUX7yljOv0hXttXz06g6bV8nMq2wTxD+7L2NMjqkqLx8Z5ueHh09NuverG9vxenJZ38uYM2X7JPW3ch2IMcZJDj85MMjOo6OMTMa4dlUjH7zSJt0z+XHOBCEi/0dVPyYibzA9eukUVV2fs8iMWWSSKeX5N0/yRs84Y6E471rdxDabdM/k0fnuID7nvt+a60CMWcxmTrr3Xpt0zxSAczZqqurUlN6fVtVj6S/g07kPz5jiF4kn2f5qD/v6JgjFkty8tsWSgykI2fZ63Zhh2ba5DMSYxSgUS/Dkrm4ODQSJJ5TbrmzjXRcvyXdYxgDn74P4HZw7hVUi8nraqmrg33IZmDHFbiISZ/uuHk6MhBARPnJ1O2uW2qR7pnCcrw/iH4HvA/8DeDBtecDmYTLmwo1MxnhqV/epSffuuLqDVUuq8h2WMac5Z4JQ1XFgHLhrfsIxpvgNTER46tUe+sbC1FaUcuemZTbpnilI9sy+MfPo1KR742Faqsu5a1MnTdU26Z4pTJYgjJknhweDPPt6H73jYZY3+Llrcye1NumeKWCWIIzJIVVlLBTnte4xfn54mLFwnNXNzqR7lWX2v58pbPYXakyOqCr7TwZ45egIrx4fYywU44qltdyzeRnlNiOrWQDsr9SYHDkxEuJ7r/fRMxZmPBTj4uZqNq9qJBxPUe7Ld3TGnJ8lCGPmWCql7O4e4wd7+jgxEiaZUtYsrWHLRU2UiDAZS1DvtwxhCp8lCGPmUP9EhB/t66d3NMxQIEZtRSnXrGhgaV3FqW381rxkFgj7SzVmDsQSKX52eIjdx8cYnozi83q4cU0Lyxv9jIfjp7Zr8Puoq7SRS2ZhsARhzDt0eDDIi27lt0AkwaUt1XxgfRtN1WWnRjFNxhL4fV7qKktt+m6zYFiCMOYCBSJxfrJ/kAP9AYYCURr8Zdy+oY1NKxtPJQERod7vsz4HsyBZgjBmllIp5fWecf7t0BAjk1FiCeXq5Q1sW9uKv9z+lzLFw/6ajZmFgUCEF/YN0D0aYiQYo6O+khvXtHBJa3W+QzNmzuU0QYjI+4G/AjzAN1X1izPWi7v+FiAE/Lqq7nLXHQUCQBJIqGpXLmM15lxiiRS/eHuYXcdGGQrG8HlKePclS7hpTQteT7ZlVYxZWHKWIETEA3wVp9hQN/CKiDytqm+mbbYNWO2+NgNfd9+nbFXVoVzFaEw23h6a5MdvDTA4EWEikmB1SxUfWNdGc015vkMzJqdyeQexCTikqkcAROTbwO1AeoK4Hfh7VVXg5yJSJyJtaaVOjcmbYDTBT/cPsv/kBIOBKPV+Hx+8so1rVzXaSCSzKOQyQbQDJ9K+d3P63cHZtmkH+gAFnhMRBb6hqo9k+iUich9wH0BnZ+fcRG4WNVXljZ5x/vXQEMPBGNFEio2d9XxgXZt1QptFJZd/7ZkusXQW21yvqr0i0gw8LyJvqepLZ2zsJI5HALq6umb+fGNmZSgY5YV9/RwfcTqhl9ZV8L41zVzeZqVAzeKTywTRDSxL+94B9Ga7japOvQ+IyHacJqszEoQxcyGeTPHLt0fYcXSU4ckoHhGuv7iJm65owef15Ds8Y/IilwniFWC1iKwEeoA7gbtnbPM08Bm3f2IzMK6qfSLiB0pUNeB+vgn4sxzGahaxY8OTvLBvgIFAhIlwgoubq9i2rpW22orz72xMEctZglDVhIh8BvghzjDXR1V1r4h8yl3/MPAszhDXQzjDXH/D3b0F2O52BHqBf1TVH+QqVrM4hWJOJ/S+vgkGg1FqK3xsW9fKuy5usk5oYwBxBhAVh66uLt2xY0e+wzAFTlXZ2zvBvxwcYjgYJRJPsra9llvWtVFjJUDNIiMiO8/2nJkNyTCLynAwygtvDXB8eJLhYIzW2go+fFU7a5ZaJ7QxM1mCMItCIpnil0fdTuhglBIRrruoiZvXWie0MWdjCcIUvRMjIV7Y10//RISxUJyLm6t4/9pW2usr8x2aMQXNEoQpWuFYkpcODrK3Z9zphC4v5ea1rbxn9RJKSqwT2pjzsQRhio6q8mbfdCd0OJZkfXsdt6xvo9Y6oY3JmiUIU1RGJ2O88NYAR4eCDAdjtNSUc9uV7azrsE5oY2bLEoQpColkih3HRnnl7RGGJ6OAsOWiRm6+opWyUuuENkVKFcKjEAuCrwoq6mEOn+GxBGEWvO7REC/sG+DkRISxyRgrl1SxbW0ryxqsE9oUsVgIjv8cxo5BZALaroSqZmi+fM6ShCUIs2BF4kn+5eAQb3SPMRSMUVXm5cYrWnjvJc3WCW2KgypEJyA0DJPDzntoyP0+BBO9kIxBKgENK6HE49xRVDbMya+3BGEWHFXlrZMBXjowyFAwSiiWZO3SGrata6XeX5bv8IyZvVTSObFPnfhDw9OvRNR9RZzXlETEuVOobHSal8Q9nceCliDM4qGqjIXiTMYSJJLKzmOjvD00yXAwSnN1OR9Y18aGzvp8h2nM+SWi0yf+U4lgxEkOyZibBKLO5ymlFVBeC9UtzntNu3O34K2AoQNn/g5f1ZyFawnCFDRV5UB/kKFAlAP9Ad7smyCeTFFbUcrmVY28f20L5aX2Z2wKiCrEJtOagtLuBqITkIxP3w2kks4+Is6JvbwWKuqgogFql0H9cvA3Q0mGuueq0z93SmWj01E9R+z/LFOwxsNx9vSM84sjwwwEnOcZApE4rbXlfOTqDtZ31OU7RLOYpVIQGTu9OWjqriAegaR7NxCPcKoOWonHSQJVzc67vxnqOqFuOVTMcii2iNMhbaOYzGIQTSTpHg1zfDjEseFJRiZj9AciDE5EAfCXednYWc9VnfU0+H15jtYsGsm40wwUGjo9EYRHIZHeLBSd3sfrg/I68C9xm4fa3USwDErL5y42Eae/YY76HGayBGHyJpVSBoNRjg5NcmwkRN9YhHA8QTCaIJlUSr0lLKkqp6GyjIuW+Gmrq6DEvTry++xP18yxWCjtbmDISQqTQ06z0KlEEHFGDE3x+Z1EUNvhNA3VLXPuBqpawbPw/0YX/hGYBSUQiXNsOMTxEecVjMSZjCaJxJOICI1VPta01rC6pYorltZS4fNwoD/IyOR0p12D30ddpU2ZYS6AKkTGpzuH0+8KYpOnjxjSlLOPlEBZtXOVXlEHlU3TzUKVDXPapFNoLEGYnIolUvSMhTk2PMnxkRBDgSjheJJgxLkK85d56aivYFljJRs66miuOfP2+5KWqlOjmPw+L3WVpVbxzZxbKukmgBnPDoSGT78bSKQ1C3lK3VFCbc5dQVWrmwg6ocyft0PJJ0sQZk6pOs1GTj9CiJ6xsJsQ4iSSitdTQktNOZe11rB2aS0XNfvxejKM0EgjItT7fdRbv4OZaWrY6MxnB8JjacNGI04/wpRTw0ZbnTuCmg5ntFB1O3jtzjSdJQjzjk1GE26zkXOXEIg4/QiRmNNs1OD3cWlrDRctqWJ9Ry3+MvuzM7Og6ozSmTlkNDQE0YCbCKKZh41W1DnJoLLRGTZat9wZQWR3oFmx/1PNWaU/oJbetJNIpugdi3BsZJKjwyGGAhFnCGo0AQqVPi9Laytor69gw7I62mrLrUnInF/6sNGZdwTxsDNKKD7VLHS2YaMtTpNQ/XIor8nn0RQFSxAmo6kH1EYmY6gqE5EE4VgSRekdCxOKJQlEEiSSKTyeElqqy1jdUs2apTVc0lxFqZXxNJB5ttFUIsPTxMMzho1GTn+a2Fs2/dzA1NPE9cud0UNem14lVyxBmIzGQnHeHprk7aFJekbDTMYSROJJKko9lHlLqPf7WN1cxcolfq7sqKWmwvoHTJpE1Bkt1PMqTHQ7TUGRCSc5lJa7TxNHMwwbdZ8mrutwkkltB9StcIeN2kXHfLMEYU4TiSc50B/gXw8OcWggyGQsQTyRwuf1uH0JVdxwWTMd9ZXWbLRYpVLOHUF0wjnpn/Y+5nxORCASgECfkwBSceduApz+gKol4G+cfphsalqJOX4S2LwzliAMqkr3aJi9vRMcGggQiCQYDEQJRhO01JSzssnPxc1+Sj0eLmutttFExS4Rc0/245mTQDTg3AEkY+4rfnpzEECJFzQJvkpnUrnSCueZgaoWaO+C5kvzc2xmVixBLGKBSJw3eyd4s2+C4WCM8XCMRFJpqSnnVy5dQltNOUmd3t4eUCsCUxPJTSWAUyf/tGQQD0EycXoC0OTpP6e0wn2KeInz7vM7HcRVS5zmIH+T8/P6954ZQ9WS+TlW845ZglhkkinlyGCQvb0THB2aJBhNMBlNUOnzcmlrDV0r6rm0pRoROesoJlPAkokzT/in3QkEzrzyP+Pq3+Oc8Mv84Gt2+wXqnJFCVS3OK5sRQhX1TnNSDmcbNbllCWKRGApG2ds7wb6+CSbCcUZDMUpE6Kiv5LqLG7l2ZSOVM55PsAfUCoyqc3V/WpPP+OlJIBZy2vvTT/6pGVf/3nInAVQ2OCd/n9/pB6hqdh4eq1wyNw+MzcNsoya3LEEUsUg8ycH+IHt6x+kbCxOIOCORait9XNVZz5aLGumot7rNOTWbovKp5Nnb/aeSQSL9yn+q7T+tHVBK0pp83Oaf8lo3AbROF52Zr5N0jmcbNbllCaLIZOpwngjHKfWUsLzRz7qOWjYuq7XnFOaDKgzsc5pYVJ2TubfMuVKPBs5MBrHJMzt/04eAgrP/qat/NxFUNrlt/23Ouz0XYOaIJYgFLL2PQBV6x8JndDg315RzxdIa3rW6iQar1zy3UilIhJ1mnbj7ioUgPuk8+Rvsh6FDzpDPeHi6uHx5nVMhLBmbHvoJztV/aSWUVTmdvD4/lNVCVRP43at/a6Ix88gSxAKVTKbYdXyMI0PO/EcnxyMois9Tgn+qw3l5PZe2VlvHcrZUnRN5POyc5KdO9rHQjGVTr7DTLJSKO+/J+PR3nXpWIDj98z1ed8hnOdS2OwmgotF5Ori62WkCmstiMsa8Q5YgFoBIPMlgIMpQMMpQMMZQMEr3SIjesTCxZIpwPIUAS6rL2LKqgRvXtJ7R4bygzaYdf+Z+U1fvsUn3Ct+9uj/1Of3KP+w+1DXjlUycOcxzirfM6fQtrXDiKi13v/udO4LQiHNHUF7r1BSQEmi5wtrkzYJQRGeRheNsw0dTKWU0FGMwGGUoEHMTQpRAJE40kSISTxGNOyeqYCxBIqlUlXnpqPexZmkNzdXlLGuoKL7kkN6On0o4J+PajukT/cymnVMn/rBzck+lXdmfOvGf7YTvc67yveXOyb/U/Vxa6RaUr4XyqcIxDedu70+PfYoN8zQLSBGdSRYGVWX/yQC9YxEmInHGw3FSKcXnLWFkMkY0kSQSTxGJJ0mmnPZpj6eEuopSOurLqKvwsayhgobKUgaDsTOaj+a0FOeFXrln+jmphFu3NzZdvzcRm/Eene6gnfocGoGx49Odtyn3ar6iwXlaN/1KPxOPz72qLwNvjXuyd0/4ZTVOf0BlvfNe0QC+infyL3Y6G+ZpFrhFnyDO9TBYNg+KnW2bSDzJRNhJAFOJYDwc5+R4hCODkyRSSiKZIpZMEU+kqKv0UeYtwV/mpa7SR31lKS01ZXQ2+umsr6Tc5znj96Y0wMToACXxEKnSSmrqm+fuSWdVOLnH6WhNJZwTdFmVM5Vy+gn8bCf7ZOz0ZZp0OnU16Vy961k+zzSzHb/E61T+0pRzYve6TTreCufhrYo6KK93rvQrG51EkM8Tsg3zNAtYThOEiLwf+CvAA3xTVb84Y724628BQsCvq+qubPadC6rKgZNnnmQvaa0GyLhu5RI/YfcKPxRN8FZfgJHRISLhMIGkl6TXT4XPQzSRJJZU4onpJABO09BkJEE5Maq9cZrKK6ioqmVtRy2bVjTQWFWWVaeyAJeUnCAoA0RKkpSLhypCSPxipyklMTVUMjp99Z1x2dTnqav3+PRMnIE+J1FoavpVUe9claef8Kdq955LicfZz1MKJaXgKZ/+fmqZb3reHl+Fc3JPJp3ZQEvd5d4K56Rr7fjG5FzOEoSIeICvAjcC3cArIvK0qr6Zttk2YLX72gx8Hdic5b7v2NhkjMHDr3K8r5+kCikg7vWzt3UlSVUGe49BPEQ8CdGkEpEy/HUteL0lpFIpQpE4wdGTSDyMqiKaJOXx4a+px19aQp1P8JcK/nLwlwqNFR7KS5TE0GGqdQLBOfkm8dPCKqr7PGlX1WlX11Pf0z9HA8h4D9WaojoVn+5IrWxy2tFTqQxX6anpE3v6w1WZxIJOW36J1zm5T125e8uch66mTuwen9uMU+m+3Dl6Sv3ORG0+v9O04i278OYpa8c3Ji9yeQexCTikqkcAROTbwO1A+kn+duDvVVWBn4tInYi0ASuy2PcdC08M4hk/TvOJHxOPT7dhy/FKQKiKTU5vPFXAasxPRVkZnhJB41EkNoEHxVsC5R6lXKDG20RTTRUlgtv+rhBJQRg0HiI42u/eUSiglHpKqAq/PrsHnGY2vcD0Cbii3j2h+8BT4V6dp12pl3jddnn3Kt1b7l6x+90TfJUz+mf4sJMc0s33lbu14xuTN7lMEO3AibTv3Th3Cefbpj3LfQEQkfuA+wA6OztnFaCfKJWlJVRWl5EKRxCgRJSKyhQlArFQjBJRvAKlHqG0BOrrqiiv9oNAJBgnOOFkDkVAPKgI1RVllFTUOEMaxeOcZKUEpASJjFPlqySSgERK8Xq8lJeVIlOToJV4p1+eqc+lad/dE31sEkaPgLjLSsvBUwZt6+bmBK7qPO1bCFfu1o5vTF7kMkFkusSb2a5xtm2y2ddZqPoI8AhAV1fXedpNTldTW0d9YxMB/wdPLasu97Lssi4ATry1wymrCUTcdWWXdTmFToCyyWEG3G3S91+Sts0ZQiNI/17OGCsz2ytzVaf5JlcncLtyN2bRy2WC6AaWpX3vAHqz3MaXxb7vmFQ2sKyjg+DIAJFEknKvh6qGZsQ9UZ9rXTb7ZzRXUyDPxwncrtyNWdRymSBeAVaLyEqgB7gTuHvGNk8Dn3H7GDYD46raJyKDWez7zokgzWuorm6jOsNJ9lzrstn/bL9zzk7sdgI3xuRQzhKEqiZE5DPAD3GGqj6qqntF5FPu+oeBZ3GGuB7CGeb6G+faNyeBnuskm80J+EJO0nZiN8YsAKI6q2b7gtbV1aU7duzIdxjGGLNgiMhOVe3KtK5kvoMxxhizMFiCMMYYk5ElCGOMMRlZgjDGGJNRUXVSu8Njj+U7jhxoAobyHUSOLYZjhMVxnHaMC8tyVV2SaUVRJYhiJSI7zjbKoFgshmOExXGcdozFw5qYjDHGZGQJwhhjTEaWIBaGR/IdwDxYDMcIi+M47RiLhPVBGGOMycjuIIwxxmRkCcIYY0xGliDyQEQeFZEBEdmTtqxBRJ4XkYPue33auodE5JCI7BeRm9OWXy0ib7jrviJSONV8RGSZiLwoIvtEZK+IfM5dXmzHWS4ivxSR19zj/C/u8qI6TnDqzIvIqyLyPfd7UR2jiBx1Y9stIjvcZUV1jLOmqvaa5xfwHuAqYE/asv8JPOh+fhD4C/fzGuA1oAxYCRwGPO66XwJbcCrwfR/Ylu9jSzueNuAq93M1cMA9lmI7TgGq3M+lwC+Aa4vtON34fh/4R+B7Rfo3exRomrGsqI5xti+7g8gDVX0JGJmx+HbgW+7nbwEfSlv+bVWNqurbOLUzNolIG1Cjqi+r81f592n75J2q9qnqLvdzANiHU2u82I5TVTXofi11X0qRHaeIdAAfAL6ZtriojvEsFsMxnpUliMLRoqp94JxcgWZ3eTtwIm27bndZu/t55vKCIyIrgI04V9dFd5xu08tuYAB4XlWL8Tj/EngASKUtK7ZjVOA5EdkpIve5y4rtGGcllyVHzdzI1H6p51heUESkCngS+D1VnThHc+yCPU5VTQIbRKQO2C4ia8+x+YI7ThG5FRhQ1Z0i8t5sdsmwrKCP0XW9qvaKSDPwvIi8dY5tF+oxzordQRSOfvf2FPd9wF3eDSxL264D6HWXd2RYXjBEpBQnOfyDqj7lLi6645yiqmPAT4D3U1zHeT1wm4gcBb4N3CAij1Fcx4iq9rrvA8B2YBNFdoyzZQmicDwN3Ot+vhf4TtryO0WkTERWAquBX7q3uwERudYdJfHJtH3yzo3pb4B9qvrltFXFdpxL3DsHRKQCeB/wFkV0nKr6kKp2qOoK4E7gx6r6cYroGEXELyLVU5+Bm4A9FNExXpB895Ivxhfw/wN9QBzniuM3gUbgBeCg+96Qtv0f44yS2E/aiAigC+eP+DDw/+I+GV8IL+BdOLfWrwO73dctRXic64FX3ePcA3zBXV5Ux5kW43uZHsVUNMcIrMIZlfQasBf442I7xgt52VQbxhhjMrImJmOMMRlZgjDGGJORJQhjjDEZWYIwxhiTkSUIY4wxGVmCMMYYk5ElCGOMMRlZgjAmh0RkhTg1Mf7arRfxnPvEtTEFzxKEMbm3Gviqql4BjAEfyW84xmTHEoQxufe2qu52P+8EVuQvFGOyZwnCmNyLpn1OYtPsmwXCEoQxxpiMLEEYY4zJyGZzNcYYk5HdQRhjjMnIEoQxxpiMLEEYY4zJyBKEMcaYjCxBGGOMycgShDHGmIwsQRhjjMno/wLoG2MPvn4UXAAAAABJRU5ErkJggg==
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<pre>fastest is 6.8440x faster than faster (<span class="ansi-green-fg">p=0.03783</span>, mean: faster: 0.0477374, fastest: 0.0069751)
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<h2 id="Closing">Closing<a class="anchor-link" href="#Closing">¶</a></h2><p>Be careful translating math into code. Also, if you have a <em>symmetric</em> matrix, such as a covariance or Kernel matrix, it is generally more efficient (in terms of constant factors) and numerically stable to perform rank-one updates to the Cholesky factorization.</p>
<p><strong>Acknowledgements</strong>: I'd like to thank <a href="http://lowrank.net/nikos/index.html">Nikos Karampatziakis</a> and <a href="https://suzyahyah.github.io/about/">Suzanna Sia</a> for comments, corrections, and/or suggestions that improved this article.</p>
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</script>On the Distribution of the Smallest Indices2021-03-20T00:00:00-04:002021-03-20T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2021-03-20:/blog/post/2021/03/20/on-the-distribution-of-the-smallest-indices/<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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<p><a href="https://timvieira.github.io/blog/post/2021/03/18/on-the-distribution-functions-of-order-statistics/">Earlier</a>, I wrote about the distribution functions of order statistics. Today, we will look at the distribution of rankings, $R_{(1)}, \ldots, R_{(n)}$: the indices of the smallest to largest items. Let $Z_{1}, \ldots, Z_{n}$ be independent random variables distribution functions $F_1, \ldots F_n$ and density functions $f_1, \ldots f_n$.</p>
<p>The ranking is generated as follows:
$$
\begin{align*}
& Z_{1} \sim F_1; \ldots; Z_{n} \sim F_n \\
& R_{(1)}, \ldots, R_{(n)} = \mathrm{argsort}\left( Z_{1}, \ldots, Z_{n} \right)
\end{align*}
$$</p>
<p>We will look at</p>
<ul>
<li><p>the probability mass function for the (unordered) set of $K$ smallest elements, $Y_{(K)} = \{ R_{(1)}, \ldots, R_{(K)} \}$</p>
</li>
<li><p>the probability that index $i$ is included in $Y_{(K)}$</p>
</li>
</ul>
<p>I care about the inclusion probabilities because they are crucial to estimation under sampling without replacement designs <a href="https://timvieira.github.io/blog/post/2017/07/03/estimating-means-in-a-finite-universe/">(Vieira, 2017)</a>. Previously, I was only able to (efficiently) use noisy estimates of the inclusion probabilities. This post develops an algorithm for efficiently approximating the inclusion probabilities of ordered sampling without replacement designs. The algorithm will require a one-dimensional numerical quadrature, which is preferable to summing over permutations.</p>
<p>The probability of the set $R_{(k)}$ is useful for other estimators, e.g., <a href="https://arxiv.org/abs/2002.06043">Kool+ (2020)</a>.</p>
<p>As with yesterday's post, I'd like to acknowledge that my derivation and implementation draws heavily on <a href="https://link.springer.com/article/10.1023/A:1010091628740">Aires (1999)</a> and <a href="https://www.sciencedirect.com/science/article/abs/pii/S0378375803001502">Traat+ (2004)</a>.</p>
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<h2 id="PMF-of-$Y_{(K)}$-.">PMF of $Y_{(K)}$ .<a class="anchor-link" href="#PMF-of-$Y_{(K)}$-.">¶</a></h2>
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<p>We can also compute the probability mass function for the entire set.
$$
\begin{align*}
\mathrm{Pr}\left( Y_{(k)} = y \right)
&=
\int_t \sum_{j}
\mathrm{Pr}\left( Z_j = t, R_{(k)} = j, Y_{(k)} = y\right) \\
&=
\int_{t}
\sum_{j \in y}
f_j(t)
\prod_{i \in y: i \ne j} F_i(t)
\prod_{i \in \overline{y}: i \ne j} (1-F_i(t))
\end{align*}
$$</p>
<p>The sum $j \in y$ partitions the event space by which index $j$ is the is the $k^{\mathrm{th}}$ smallest element, $R_{(k)} = j$. Now, with $j$ is fixed, we integrate over the density of values $t$ that $Z_j$ can take on. Since $j$ is decided as the $k^{\mathrm{th}}$, it must be in the set. Additionally, we don't want to multiply the probability of $t$ twice, so we exclude $j$ for the two products. $\prod_{i \in y: i \ne j} F_i(t)$ is the probability that all $i \in y$ that aren't $i=j$ are larger than the threshold $t$. It is a product because they are independently drawn. Similarly, the other product is the probability that $i \in \overline{y}: i \ne j$ are all smaller.</p>
<p>As a minor touch up on this formula to simplify implementation, we apply a change of variables $t=F^{-1}_j(u)$ to integrate over $u \in [0, 1]$ instead of $t \in \mathrm{domain}(Z_i)$. The change of variables formula means that we end up dividing by the density because of the usual "determinant of the Jacobian thing," $\frac{d t }{d u} = \frac{d}{d u} F^{-1}_j(u) = \frac{1}{f_j(t)}$ by the <a href="https://en.wikipedia.org/wiki/Inverse_function_theorem">inverse function theorem</a>). Our final result is
$$
\int_0^1
\sum_{j \in y}
\prod_{i \in y: i \ne j} F_i(F^{-1}_j(u)) \prod_{i \in \overline{y}: i \ne j} (1-F_i(F^{-1}_j(u))) \, du
$$</p>
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<p>Although this equation involves an integral, it is a massive speedup over summing permutations of the indices. I should say that in some exceptional cases, the integral may simplify further. This equation is implemented and tested in the Code and Tests sections.</p>
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<p>Lastly, I'd like to give a shout-out to Dan Marthaler, a helpful commenter on this blog, who pointed out that the PMF of the multivariate noncentral hypergeometric distribution would help evaluate the PMF of the Gumbel sorting scheme. I'd also like to point out that Appendix B of <a href="https://arxiv.org/abs/2002.06043">Kool+2020</a> gives a very similar derivation for the PMF of the Gumbel sorting scheme. My version gives a generic recipe for any collection of independent variables, not just Gumbels. The general case does not have closed-form integrals, so using numerical integration may be the best we can do.</p>
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<h2 id="Inclusion-probabilities">Inclusion probabilities<a class="anchor-link" href="#Inclusion-probabilities">¶</a></h2>
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<p>We can also ask for the inclusion probability of $i$ in the bottom $k$ (i.e., the set of $k$ smallest indices). Let $Y_{(k)} = \{ R_{(1)}, \ldots, R_{(n)} \}$.
$$
\mathrm{Pr}\left( i \in Y_{(k)} \right)
$$</p>
<p>Given the distribution function for the order statistics that we worked out <a href="https://timvieira.github.io/blog/post/2021/03/18/on-the-distribution-functions-of-order-statistics/">yesterday</a>, the inclusion probability of each item can be computed by the following integral (<a href="https://link.springer.com/article/10.1023/A:1010091628740">Aires, 1999</a>, Eq. 6).</p>
$$
\begin{align*}
\mathrm{Pr}\left( i \in Y_{(k)} \right)
&= \int_{t} f_i(t) \, \mathrm{Pr}\left[ \underset{{j \in [n]\smallsetminus\{i\}}}{\overset{(k-1)}{\mathrm{smallest}}}\ Z_j > t \right] \\
&= \int_{t} f_i(t) \, (1-F^{[n] \smallsetminus \{i\}}_{(k-1)}(t))
\end{align*}
$$<p>where $F^{[n] \smallsetminus \{i\}}_{(k-1)}$ is the distribution function of the $(k-1)^{\mathrm{th}}$ order statistic with index $i$ removed from the set $[n]\overset{\mathrm{def}}{=}\{1,\ldots,n\}$. The reason for this removal is that have have forced $i$ to be in $Y_{(k)}$, so we're only sampling from what remains; namely, $(k-1)$ from the set of remaining elements $[n] \smallsetminus \{i\}$.</p>
<p>Now, integrating over the density of values $t$ of $Z_i$, we will cover all the ways that $i$ can be the $k^{\mathrm{th}}$ smallest element.</p>
<p>As a final step, we apply a change of variables $t = F^{-1}_i(u)$ to make the integral over $u$ from $0$ to $1$ instead of t, which ranges over the entire domain of $Z_i$. (See the earlier explanation for with the pdf disappears.)
$$
= 1 - \int_0^1 F^{[n] \smallsetminus \{i\}}_{(k-1)}(F^{-1}_i(u)) \, du
$$</p>
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<h2 id="Code">Code<a class="anchor-link" href="#Code">¶</a></h2>
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<p>Code from <a href="https://timvieira.github.io/blog/post/2021/03/18/on-the-distribution-functions-of-order-statistics/">order statistics post</a>.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">numba</span> <span class="kn">import</span> <span class="n">njit</span>
<span class="k">class</span> <span class="nc">Ordered</span><span class="p">:</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ds</span><span class="p">:</span> <span class="s1">'list of distributions'</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">ds</span> <span class="o">=</span> <span class="n">ds</span>
<span class="k">def</span> <span class="nf">rvs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sample_keys</span><span class="p">(</span><span class="n">size</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">sample_keys</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">d</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">])</span>
<span class="k">def</span> <span class="nf">cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
<span class="s2">"Evaluate the CDF of the order statistics."</span>
<span class="k">return</span> <span class="n">C</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">]))[</span><span class="mi">1</span><span class="p">:]</span>
<span class="k">def</span> <span class="nf">pdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
<span class="s2">"Evaluate the PDF of the order statistics."</span>
<span class="c1"># We implement a more efficient expression than we derived above</span>
<span class="c1"># by manually applying algorithmic differentiation.</span>
<span class="n">_</span><span class="p">,</span> <span class="n">d_p</span> <span class="o">=</span> <span class="n">d_C</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">]),</span> <span class="n">K</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># off-by-one due to shift in cdf</span>
<span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">d_p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">[</span><span class="n">n</span><span class="p">]</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">)))</span>
<span class="nd">@njit</span>
<span class="k">def</span> <span class="nf">B</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="s2">"Sum of weighted sets of size == k."</span>
<span class="n">q</span> <span class="o">=</span> <span class="mi">1</span><span class="o">-</span><span class="n">p</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="n">F</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">return</span> <span class="n">F</span><span class="p">[</span><span class="n">N</span><span class="p">,:]</span>
<span class="nd">@njit</span>
<span class="k">def</span> <span class="nf">C</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="s2">"Sum of weighted sets of size >= k."</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">B</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
<span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
<span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">b</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<span class="k">return</span> <span class="n">c</span>
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<p>New code</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy.integrate</span> <span class="kn">import</span> <span class="n">quadrature</span>
<span class="k">class</span> <span class="nc">Ranks</span><span class="p">:</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ds</span><span class="p">:</span> <span class="s1">'list of distributions'</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">ds</span> <span class="o">=</span> <span class="n">ds</span>
<span class="k">def</span> <span class="nf">sample_indices</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sample_keys</span><span class="p">(</span><span class="n">size</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="k">def</span> <span class="nf">sample_keys</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">d</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">])</span>
<span class="k">def</span> <span class="nf">inclusion</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
<span class="s2">"Pr[ i \in Y_{(K)} ]"</span>
<span class="k">if</span> <span class="n">K</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">):</span> <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">))</span>
<span class="n">target</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">ppf</span>
<span class="n">ds</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">);</span> <span class="n">ds</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="c1"># remove i from the ensemble</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">Ordered</span><span class="p">(</span><span class="n">ds</span><span class="p">)</span><span class="o">.</span><span class="n">cdf</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">-</span><span class="n">quadrature</span><span class="p">(</span><span class="k">lambda</span> <span class="n">u</span><span class="p">:</span> <span class="n">F</span><span class="p">(</span><span class="n">target</span><span class="p">(</span><span class="n">u</span><span class="p">))[</span><span class="n">K</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span>
<span class="n">maxiter</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span> <span class="n">vec_func</span><span class="o">=</span><span class="kc">False</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">pmf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="s2">"Pr[ Y_{(K)} = y ]"</span>
<span class="k">return</span> <span class="n">quadrature</span><span class="p">(</span><span class="k">lambda</span> <span class="n">u</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pmf</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">200</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">_pmf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
<span class="n">c</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">ds</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">ds</span><span class="p">)</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">y</span><span class="p">:</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">ds</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="n">z</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span> <span class="k">pass</span>
<span class="k">elif</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">y</span><span class="p">:</span>
<span class="n">z</span> <span class="o">*=</span> <span class="n">ds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">z</span> <span class="o">*=</span> <span class="n">ds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">sf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="n">c</span> <span class="o">+=</span> <span class="n">z</span>
<span class="k">return</span> <span class="n">c</span>
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<h2 id="Tests">Tests<a class="anchor-link" href="#Tests">¶</a></h2><p>Below, we test out our analytical method for computing the distribution function of independent normal distributions. The method should work for any collection of distributions in <code>scipy.stats</code>. You can even mix and match the distribution types. We will compare the analytical $F_{(k)}$ to the empirical distribution of $Z_{(k)}$ using a large sample size.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.stats</span> <span class="k">as</span> <span class="nn">st</span><span class="o">,</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span><span class="o">,</span> <span class="nn">pylab</span> <span class="k">as</span> <span class="nn">pl</span>
<span class="kn">from</span> <span class="nn">numpy.random</span> <span class="kn">import</span> <span class="n">uniform</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">123456</span><span class="p">)</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">8</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">Ranks</span><span class="p">([</span><span class="n">st</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)])</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">m</span><span class="o">.</span><span class="n">ds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">ts</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="n">i</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">);</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal</span> <span class="kn">import</span> <span class="n">ok</span><span class="p">,</span> <span class="n">warn</span>
<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">Counter</span>
<span class="n">reps</span> <span class="o">=</span> <span class="mi">10_000</span>
<span class="n">tol</span> <span class="o">=</span> <span class="n">reps</span><span class="o">**-</span><span class="mf">0.5</span>
<span class="n">K</span> <span class="o">=</span> <span class="mi">3</span>
<span class="c1">#K = int(np.random.randint(1, N+1))</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'N=</span><span class="si">{</span><span class="n">N</span><span class="si">}</span><span class="s1">, K=</span><span class="si">{</span><span class="n">K</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
<span class="n">pmf_</span> <span class="o">=</span> <span class="n">Counter</span><span class="p">()</span>
<span class="n">incl_</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">Rs</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">sample_indices</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
<span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">reps</span><span class="p">):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">x</span><span class="p">[</span><span class="n">Rs</span><span class="p">[</span><span class="n">r</span><span class="p">,</span> <span class="p">:</span><span class="n">K</span><span class="p">]]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">incl_</span> <span class="o">+=</span> <span class="n">x</span> <span class="o">/</span> <span class="n">reps</span>
<span class="n">pmf_</span><span class="p">[</span><span class="nb">frozenset</span><span class="p">(</span><span class="n">Rs</span><span class="p">[</span><span class="n">r</span><span class="p">,</span> <span class="p">:</span><span class="n">K</span><span class="p">])]</span> <span class="o">+=</span> <span class="mi">1</span><span class="o">/</span><span class="n">reps</span>
<span class="c1"># sanity check: incl probs for a size K sample should sum to K</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">incl_</span><span class="o">.</span><span class="n">sum</span><span class="p">(),</span> <span class="n">K</span><span class="p">)</span>
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<pre>N=8, K=3
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<p>Below, we compare Monte Carlo inclusion probabilities to our numerical integration approach.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="o">-</span><span class="n">incl_</span><span class="p">):</span>
<span class="n">have</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">inclusion</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">i</span><span class="si">:</span><span class="s1">-2d</span><span class="si">}</span><span class="s1">: </span><span class="si">{</span><span class="n">have</span><span class="si">:</span><span class="s1">.4f</span><span class="si">}</span><span class="s1">, </span><span class="si">{</span><span class="n">incl_</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="si">:</span><span class="s1">.4f</span><span class="si">}</span><span class="s1">'</span><span class="p">,</span>
<span class="n">ok</span> <span class="k">if</span> <span class="mf">0.5</span><span class="o">*</span><span class="nb">abs</span><span class="p">(</span><span class="n">have</span> <span class="o">-</span> <span class="n">incl_</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o"><=</span> <span class="n">tol</span> <span class="k">else</span> <span class="n">warn</span><span class="p">)</span>
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<pre> 4: 0.6408, 0.6454 <span class="ansi-green-fg">ok</span>
6: 0.5872, 0.5864 <span class="ansi-green-fg">ok</span>
0: 0.5799, 0.5746 <span class="ansi-green-fg">ok</span>
1: 0.4595, 0.4570 <span class="ansi-green-fg">ok</span>
3: 0.2776, 0.2790 <span class="ansi-green-fg">ok</span>
5: 0.2489, 0.2479 <span class="ansi-green-fg">ok</span>
2: 0.2040, 0.2071 <span class="ansi-green-fg">ok</span>
7: 0.0021, 0.0026 <span class="ansi-green-fg">ok</span>
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<p>Below, we compare the Monte Carlo estimate of the PMF to our numerical integration approach.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">for</span> <span class="n">Y</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">pmf_</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">Y</span><span class="p">:</span> <span class="o">-</span><span class="n">pmf_</span><span class="p">[</span><span class="n">Y</span><span class="p">]):</span>
<span class="n">want</span> <span class="o">=</span> <span class="n">pmf_</span><span class="p">[</span><span class="n">Y</span><span class="p">]</span>
<span class="n">have</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">pmf</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">Y</span><span class="si">}</span><span class="s1">: </span><span class="si">{</span><span class="n">have</span><span class="si">:</span><span class="s1">.4f</span><span class="si">}</span><span class="s1">, </span><span class="si">{</span><span class="n">want</span><span class="si">:</span><span class="s1">.4f</span><span class="si">}</span><span class="s1">'</span><span class="p">,</span>
<span class="n">ok</span> <span class="k">if</span> <span class="mf">0.5</span><span class="o">*</span><span class="nb">abs</span><span class="p">(</span><span class="n">have</span> <span class="o">-</span> <span class="n">want</span><span class="p">)</span> <span class="o"><=</span> <span class="n">tol</span> <span class="k">else</span> <span class="n">warn</span><span class="p">)</span>
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<pre>frozenset({0, 4, 6}): 0.1286, 0.1261 <span class="ansi-green-fg">ok</span>
frozenset({0, 1, 4}): 0.0847, 0.0851 <span class="ansi-green-fg">ok</span>
frozenset({1, 4, 6}): 0.0856, 0.0850 <span class="ansi-green-fg">ok</span>
frozenset({0, 1, 6}): 0.0783, 0.0778 <span class="ansi-green-fg">ok</span>
frozenset({0, 3, 4}): 0.0430, 0.0448 <span class="ansi-green-fg">ok</span>
frozenset({3, 4, 6}): 0.0437, 0.0437 <span class="ansi-green-fg">ok</span>
frozenset({4, 5, 6}): 0.0398, 0.0425 <span class="ansi-green-fg">ok</span>
frozenset({0, 3, 6}): 0.0390, 0.0392 <span class="ansi-green-fg">ok</span>
frozenset({0, 4, 5}): 0.0382, 0.0363 <span class="ansi-green-fg">ok</span>
frozenset({2, 4, 6}): 0.0331, 0.0354 <span class="ansi-green-fg">ok</span>
frozenset({0, 2, 4}): 0.0312, 0.0321 <span class="ansi-green-fg">ok</span>
frozenset({0, 5, 6}): 0.0301, 0.0290 <span class="ansi-green-fg">ok</span>
frozenset({1, 3, 4}): 0.0284, 0.0274 <span class="ansi-green-fg">ok</span>
frozenset({0, 1, 3}): 0.0256, 0.0249 <span class="ansi-green-fg">ok</span>
frozenset({1, 3, 6}): 0.0255, 0.0249 <span class="ansi-green-fg">ok</span>
frozenset({1, 4, 5}): 0.0255, 0.0248 <span class="ansi-green-fg">ok</span>
frozenset({0, 2, 6}): 0.0224, 0.0216 <span class="ansi-green-fg">ok</span>
frozenset({1, 2, 4}): 0.0209, 0.0215 <span class="ansi-green-fg">ok</span>
frozenset({0, 1, 5}): 0.0193, 0.0205 <span class="ansi-green-fg">ok</span>
frozenset({1, 5, 6}): 0.0198, 0.0191 <span class="ansi-green-fg">ok</span>
frozenset({1, 2, 6}): 0.0148, 0.0160 <span class="ansi-green-fg">ok</span>
frozenset({3, 4, 5}): 0.0131, 0.0147 <span class="ansi-green-fg">ok</span>
frozenset({0, 1, 2}): 0.0142, 0.0127 <span class="ansi-green-fg">ok</span>
frozenset({2, 4, 5}): 0.0131, 0.0126 <span class="ansi-green-fg">ok</span>
frozenset({2, 3, 4}): 0.0108, 0.0119 <span class="ansi-green-fg">ok</span>
frozenset({3, 5, 6}): 0.0100, 0.0098 <span class="ansi-green-fg">ok</span>
frozenset({0, 3, 5}): 0.0097, 0.0084 <span class="ansi-green-fg">ok</span>
frozenset({2, 3, 6}): 0.0076, 0.0081 <span class="ansi-green-fg">ok</span>
frozenset({0, 2, 5}): 0.0077, 0.0077 <span class="ansi-green-fg">ok</span>
frozenset({0, 2, 3}): 0.0072, 0.0075 <span class="ansi-green-fg">ok</span>
frozenset({2, 5, 6}): 0.0082, 0.0075 <span class="ansi-green-fg">ok</span>
frozenset({1, 3, 5}): 0.0064, 0.0067 <span class="ansi-green-fg">ok</span>
frozenset({1, 2, 5}): 0.0051, 0.0054 <span class="ansi-green-fg">ok</span>
frozenset({1, 2, 3}): 0.0048, 0.0045 <span class="ansi-green-fg">ok</span>
frozenset({2, 3, 5}): 0.0026, 0.0022 <span class="ansi-green-fg">ok</span>
frozenset({4, 6, 7}): 0.0003, 0.0004 <span class="ansi-green-fg">ok</span>
frozenset({4, 5, 7}): 0.0002, 0.0003 <span class="ansi-green-fg">ok</span>
frozenset({0, 1, 7}): 0.0001, 0.0003 <span class="ansi-green-fg">ok</span>
frozenset({3, 4, 7}): 0.0001, 0.0003 <span class="ansi-green-fg">ok</span>
frozenset({1, 5, 7}): 0.0000, 0.0002 <span class="ansi-green-fg">ok</span>
frozenset({0, 6, 7}): 0.0001, 0.0002 <span class="ansi-green-fg">ok</span>
frozenset({2, 4, 7}): 0.0002, 0.0002 <span class="ansi-green-fg">ok</span>
frozenset({0, 4, 7}): 0.0003, 0.0002 <span class="ansi-green-fg">ok</span>
frozenset({2, 5, 7}): 0.0000, 0.0001 <span class="ansi-green-fg">ok</span>
frozenset({1, 6, 7}): 0.0001, 0.0001 <span class="ansi-green-fg">ok</span>
frozenset({1, 4, 7}): 0.0002, 0.0001 <span class="ansi-green-fg">ok</span>
frozenset({0, 5, 7}): 0.0001, 0.0001 <span class="ansi-green-fg">ok</span>
frozenset({0, 2, 7}): 0.0001, 0.0001 <span class="ansi-green-fg">ok</span>
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<h2 id="Closing">Closing<a class="anchor-link" href="#Closing">¶</a></h2><p>This Jupyter notebook is available for <a href="https://github.com/timvieira/blog/blob/master/content/Distribution-of-Smallest-Indices.ipynb">download</a>.</p>
<p>I am very happy to answer questions! There are many ways to do that: comment at the end of this document, Tweet at <a href="https://twitter.com/xtimv">@xtimv</a>, or email tim.f.vieira@gmail.com.</p>
<p>If you found this article interesting please consider sharing it on social media. If you found this article useful please cite it</p>
<div class="highlight"><pre><span></span><span class="nc">@misc</span><span class="p">{</span><span class="nl">vieira2021smallest</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">author</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{Tim Vieira}</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">title</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{On the Distribution of the Smallest Indices}</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">year</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{2021}</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">url</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{https://timvieira.github.io/blog/post/2021/03/20/on-the-distribution-of-the-smallest-indices/}</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
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</script>On the Distribution Functions of Order Statistics2021-03-18T00:00:00-04:002021-03-18T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2021-03-18:/blog/post/2021/03/18/on-the-distribution-functions-of-order-statistics/<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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<p>Let $Z_{(1)} \le \ldots \le Z_{(n)}$ denote the <a href="https://en.wikipedia.org/wiki/Order_statistic">order statistics</a> of a collection of independent random variables, $Z_{i}$ indexed by $i \in N$ where $|N| = n$. Let $F_i$ and $f_i$ be the distribution and density functions for each $i$.</p>
<p>To generate, the order statistics we do the following:
$$
\begin{align*}
& Z_{i} \sim F_i\quad\textbf{for } i \in N \\
& Z_{(1)}, \ldots, Z_{(n)} = \mathrm{sort}\left([ Z_{i} \textbf{ for }i \in N ] \right)
\end{align*}
$$</p>
<p>In this post, we work out the distribution and density functions for each $Z_{(k)}$.</p>
<p>There is no shortage of lecture notes and books online that describe the distribution functions for $Z_{(1)}$ (min) and $Z_{(n)}$ (max) as these cases are particularly simple and more commonly used; similarly, for the case of identically distributed random variables (where the $F_i$s are equal).
The more general (non-identical, but still independent) case turns out to not be too much harder. Why do <em>I</em> care about this problem? These distribution functions arise when analyzing order-based sampling without replacement designs, such as the <a href="https://timvieira.github.io/blog/tag/gumbel.html">Gumbel-top-k design</a> that I have written about many times.</p>
<p>Before continuing, I should note that my derivation and implementation draws heavily on <a href="https://link.springer.com/article/10.1023/A:1010091628740">Aires (1999)</a>; specifically, Lemma 2.</p>
<p>Without further ado...</p>
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<h3 id="Distribution-function">Distribution function<a class="anchor-link" href="#Distribution-function">¶</a></h3>$$
\begin{align*}
F^N_{(k)}(t)
&\overset{\mathrm{def}}{=} \mathrm{Pr}\left( Z_{(k)} \le t \right) \\
&= \mathrm{Pr}\left( \text{at least } k \text{ of } Z_1 \ldots Z_n \text{ are } \le t \right) \\
&= \sum_{\substack{ Y \subseteq N \\ |Y| \ge k}}
\underbrace{\left( \prod_{i \in Y} F_i(t) \right)}_{ \text{ values } \le t}
\, \underbrace{\left( \prod_{i \in N \smallsetminus Y } \left( 1-F_i(t) \right) \right)}_{ \text{ values } > t}
\end{align*}
$$<p>It is a sum over all the ways in which $t$ can be the $k^{\text{th}}$ order statistics: there must be <em>at least</em> $k$ values that are $\le t$, and the remaining values that are $> t$. These events are mutually exclusive and exhaustive. Let $Y$ be the set of variables that are $\le t$. Clearly, we require $|Y| \ge k$. Because the $Z_i$s are independent, the probability of the event that $Y$ is the set of $k$ smallest elements is simply the product of the $F_i$s in $Y$ and $(1-F_i)$s in the complement of $Y$.</p>
<p>It turns out that we can evaluate the exponential-size sum in $\mathcal{O}(n^2)$ time with a simple algorithm, which is essentially a <em>probabilistic generalization</em> of the dynamic program for evaluating the binomial coefficients, ${n \choose k}$. Furthermore, we can simultaneously evaluate the distribution function for <em>all</em> $n$ order statistics in $\mathcal{O}(n^2)$ time. The algorithm for doing this computation efficiently is in the code section of this article.</p>
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<h3 id="Density-function">Density function<a class="anchor-link" href="#Density-function">¶</a></h3>\begin{align}
f^N_{(k)}(t)
&\overset{\mathrm{def}}{=} \frac{\partial}{\partial t} F^N_{(k)}(t) \\
&= \frac{\partial}{\partial t} \left[
\sum_{\substack{ Y \subseteq N: \\ |Y| \ge k}}
\left( \prod_{i \in Y} F_i(t) \right)
\left( \prod_{i \in N \smallsetminus Y } \left(1-F_i(t)\right) \right)
\right] \\
&=
\sum_j
\sum_{\substack{ Y \subseteq N: \\ |Y| \ge k}}
(-1)^{1[j \notin Y]} f_j(t)
\left( \prod_{\substack{i \in Y \\ i \ne j}} F_i(t) \right)
\left( \prod_{\substack{i \in N \smallsetminus Y \\ i \ne j}} \left(1-F_i(t)\right) \right) \\
&= \sum_j \left(F^{N - j}_{(k-1)}(t) - F^{N - j}_{(k)}(t)\right) f_j(t)
\end{align}<p>where
$F^{N - j}_{(k-1)}$ and $F^{N - j}_{(k)}$ are the distribution functions for the order statistics $k-1$ and $k$ over the set of elements <em>excluding</em> the random variable $Z_j$.</p>
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<h2 id="Code">Code<a class="anchor-link" href="#Code">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">class</span> <span class="nc">Ordered</span><span class="p">:</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ds</span><span class="p">:</span> <span class="s1">'list of distributions'</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">ds</span> <span class="o">=</span> <span class="n">ds</span>
<span class="k">def</span> <span class="nf">rvs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="s2">"Generate a sample of the order statistics"</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">([</span><span class="n">d</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
<span class="s2">"Evaluate the CDF of the order statistics."</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">])</span>
<span class="k">return</span> <span class="n">C</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">)[</span><span class="mi">1</span><span class="p">:]</span>
<span class="k">def</span> <span class="nf">pdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
<span class="s2">"Evaluate the PDF of the Kth order statistics."</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">])</span>
<span class="p">[</span><span class="n">_</span><span class="p">,</span> <span class="n">d_p</span><span class="p">,</span> <span class="n">d_q</span><span class="p">]</span> <span class="o">=</span> <span class="n">d_C</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># off-by-one due to shift in cdf</span>
<span class="k">return</span> <span class="nb">sum</span><span class="p">((</span><span class="n">d_p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">-</span> <span class="n">d_q</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">[</span><span class="n">n</span><span class="p">]</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ds</span><span class="p">)))</span>
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<h3 id="Utility-functions-to-efficiently-sum-over-weighted-subsets.">Utility functions to efficiently sum over weighted subsets.<a class="anchor-link" href="#Utility-functions-to-efficiently-sum-over-weighted-subsets.">¶</a></h3><p><strong>Definition</strong>: Let $p_i$ and $q_i$ be inclusion and exclusion weights (respectively) for some set of elements $i \in N$. The total weight of subsets of $N$ of size with size least $K$ is
$$
C^N_K
\overset{\mathrm{def}}{=} \sum_{\substack{Y \subseteq N:\\ |Y|\ge K}} \left(\prod_{i \in Y} p_i \right)\left(\prod_{i \in N \smallsetminus Y} q_i\right)
$$
We can evaluate $C^N_K$ efficiently in $\mathcal{O}(n^2)$ with the following algorithms.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">B</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Compute the sum of weighted sets of size == k.</span>
<span class="sd"> p and q are the inclusion and exclusion weights, respectively</span>
<span class="sd"> """</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">F</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">return</span> <span class="n">F</span><span class="p">[</span><span class="n">N</span><span class="p">,:]</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">C</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Compute the sum of weighted sets of size >= k.</span>
<span class="sd"> p and q are the inclusion and exclusion weights, respectively</span>
<span class="sd"> """</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">B</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">2</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)):</span>
<span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
<span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">b</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<span class="k">return</span> <span class="n">c</span>
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<p>The method below is used to efficiently evaluate the gradient of the <code>C</code> function in time $\mathcal{O}(n^2)$. This method is used to evaluated the PDF. It was derived by manually applying algorithmic differentiation, but it could also be evaluated using an automatic differentiation toolkit (<a href="https://timvieira.github.io/blog/post/2016/09/25/evaluating-fx-is-as-fast-as-fx/">read more</a>).</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">d_C</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
<span class="s2">"Evaluate ∇[C(p)[K]]"</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">K</span> <span class="o">></span> <span class="mi">0</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="n">F</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">2</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)):</span>
<span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">F</span><span class="p">[</span><span class="n">N</span><span class="p">,</span><span class="n">j</span><span class="p">]</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)):</span>
<span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">c</span><span class="p">[</span><span class="n">K</span><span class="p">]</span>
<span class="n">d_F</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
<span class="n">d_p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">d_q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="n">d_c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="n">d_c</span><span class="p">[</span><span class="n">K</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="n">d_c</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">d_c</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="n">d_F</span><span class="p">[</span><span class="n">N</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">d_c</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)):</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)):</span>
<span class="n">d_F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="p">]</span> <span class="o">+=</span> <span class="n">q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">d_F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">k</span><span class="p">]</span>
<span class="n">d_F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">d_F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">k</span><span class="p">]</span>
<span class="n">d_q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">+=</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="p">]</span> <span class="o">*</span> <span class="n">d_F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">k</span><span class="p">]</span>
<span class="n">d_p</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">+=</span> <span class="n">d_F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">k</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)):</span>
<span class="n">d_q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">+=</span> <span class="n">d_F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
<span class="n">d_F</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="n">q</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">*</span> <span class="n">d_F</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
<span class="k">return</span> <span class="p">[</span><span class="n">out</span><span class="p">,</span> <span class="n">d_p</span><span class="p">,</span> <span class="n">d_q</span><span class="p">]</span>
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<h2 id="Tests">Tests<a class="anchor-link" href="#Tests">¶</a></h2><p><strong>Synethic Data</strong></p>
<p>Below, we test out our analytical method for computing the distribution function of independent normal distributions. The method should work for any collection of distributions in <code>scipy.stats</code>. You can even mix and match the distribution types. We will compare the analytical $F_{(k)}$ to the empirical distribution of $Z_{(k)}$ using a large sample size.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.stats</span> <span class="k">as</span> <span class="nn">st</span><span class="o">,</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span><span class="o">,</span> <span class="nn">pylab</span> <span class="k">as</span> <span class="nn">pl</span>
<span class="kn">from</span> <span class="nn">numpy.random</span> <span class="kn">import</span> <span class="n">uniform</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">90210</span><span class="p">)</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">16</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">Ordered</span><span class="p">([</span><span class="n">st</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)])</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">m</span><span class="o">.</span><span class="n">ds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">ts</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="n">i</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">);</span>
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H594GPW2geNMS3AX4wxt8f3/bu19quVOz0RkTEY0OqsSjW/1vZpdabMr4hI7Rg2+LXWtgFt8fY+Y8w6YG6lT0xEZKzGrdVZ0L/bQ1WGFRGREoxowpsxZiFwPHB/vOtDxphHjDHfMcZMHeQxlxtj1hhj1mzfvr3YISIilZFrdeaY6rY6s/1WeNOENxGRmlFy8GuMaQZ+Bvy9tbYd+CZwGLCUKDN8dbHHWWuvs9Yus9YumzFjxtjPWESkRPnMr+tWtdVZGFocg/r8iojUoJKCX2NMiijw/ZG19ucA1tqXrLWBtTYEvg2cXLnTFBEZhVzNb67VWZUmnuUnvKnPr4hIzRk2+DXGGOB6YJ219t8K9s8uOOw84LHyn56IyOjlyxxcN8rCVnmRC2V+RURqTymZ39OBdwGvMcasjS/nAl8xxjxqjHkEeDXwkUqeqIjIiBW0OqOKrc7CeHlj1fyKiJRuz549XHjhhRx99NEsWrSIe++9tyLjlNLt4Y+AKXKXevqKSE2z/VudVXmRi3zmV63ORESG9eEPf5hzzjmH//3f/yWTydDZ2VmRcbTCm4gkVy7j6roYEwW/1lqMKfb3fPlEwS+9mV8tciEiMqT29nZ+//vf873vfQ+AdDpNOp2uyFgKfkUkuXKZX2OizC9E2V/XrdiQ1lqsRZlfEZmQ9vzyWTJbO8r6nOk5TUx502FDHvPcc88xY8YM3vOe9/Dwww9z4okncs0119DU1FTWc4ER9vkVEZlI+rQ6ywW8FZ70lkvyquZXRKR0vu/z4IMP8oEPfICHHnqIpqYmrrrqqoqMpcyviCRXmFvkIprwBvECFBUcMpflLWx1pm4PIjJRDJehrZTW1lZaW1s55ZRTALjwwgsrFvwq8ysiiZWb8Ibj5EsQKp35zdX3Rssbx/sU/IqIDGnWrFnMmzeP9evXA3DnnXdyzDHHVGQsZX5FJLkKyh7ymd8Kd3zIZXmNKezzW9EhRUQS4T/+4z+4+OKLyWQyHHrooXz3u9+tyDgKfkUksWzBCm+mcMJbRcfMlT0UdHuoUos1EZGJbOnSpaxZs6bi46jsQUSSq6DVWa4GwVZ6wlt+XQ2Tb6mmzK+ISO1Q8CsiyTVYq7NKDhkW1vxqwpuISK1R8CsiiWWD3p6++VZnVSt70IQ3EZFapOBXRJIrDKM2Z9Bb9lDFzK+jzK+ISM1R8CsiiWXDAOLgNz/hreI1v70T3vJlD4p9RURqhoJfEUmugrKHqrU6swWLXOS6PWh5YxGRmqHgV0SSyxaUPeQWuah4zW88nCa8iYiUbP369SxdujR/mTRpEl/72tcqMpb6/IpIYtkgLCh7iDO/lV7hLSic8JYre1DwKyIylKOOOoq1a9cCEAQBc+fO5bzzzqvIWMr8ikhyhUFB5rdKi1zYghXe8slmBb8iIqW68847Oeyww1iwYEFFnl+ZXxFJrPFodaZuDyIykd122228+OKLZX3OWbNm8frXv77k42+88UZWrFhR1nMopMyviCRXGPbW+lap1VnRbg9a4U1EpCSZTIZbbrmFiy66qGJjKPMrIollwwDj5DK/1Wp1Fl2r5ldEJqKRZGgr4bbbbuOEE07g4IMPrtgYyvyKSHIFYe+yxvlWZ5UNRPssb6yaXxGREVm1alVFSx5Awa+IJJkN85nf3lZn1VrkwuQnvanmV0RkeJ2dndx+++2cf/75FR1HZQ8ikljj0eqsMPjNXavmV0RkeI2NjezcubPi4yjzKyLJNQ6tzkKbK3uIbkfBrzK/IiK1QsGviCSWDW1vq7Nq9fnNZ37jjLNj8gGxiIiMPwW/IpJcQZAvP8hPeAsqnPkNeludQVRqrMyviEjtUPArIollwzAf9OZbnVV6wlsc5+YWuDCuan5FRGqJgl8RSa4gqHqrs3zZQ9znzHGMWp2JiNQQBb8ikljWhhjTW3sLVLXVGURBsMoeRERqh4JfEUmuIMxPeKNKrc4KF7kAdXsQESnFv//7v7N48WKWLFnCihUr6O7urthYCn5FJLmKtTqrcOeF3swv+WsFvyIig9uyZQtf//rXWbNmDY899hhBEHDjjTdWbDwFvyKSWMVanVV7kQtHrc5ERIbl+z5dXV34vk9nZydz5syp2Fha4U1EkisI8hPPchPeqHSrs/jpnT41vxUdUkSkbJ566l/Yt39dWZ+zpXkRRx75T4PeP3fuXP7hH/6B+fPn09DQwNlnn83ZZ59d1nMopMyviCSWDXtrfvOtziociVpbbHljZX5FRAaze/dubr75Zp5//nm2bt1KR0cHP/zhDys2njK/IpJcYdgb9ObLHqqzwpsmvInIRDRUhrZS7rjjDg455BBmzJgBwPnnn8+f/vQn3vnOd1ZkvGEzv8aYecaYu4wx64wxjxtjPhzvn2aMud0Y83R8PbUiZygiMko2DPIzz3qXN65wt4egf80v6vMrIjKE+fPnc99999HZ2Ym1ljvvvJNFixZVbLxSyh584GPW2kXAqcAHjTHHAJ8E7rTWHgHcGd8WEakdQdi7yIVbneWNe8seiK9NpRtMiIhMaKeccgoXXnghJ5xwAsceeyxhGHL55ZdXbLxhyx6stW1AW7y9zxizDpgLvAU4Iz7sBuBu4BMVOUsRkdEIQ4zTt9tDfkZaheRKinMT7bTIhYjI8K688kquvPLKqow1oglvxpiFwPHA/cDBcWCcC5BnDvKYy40xa4wxa7Zv3z7G0xURKZ0Nw97+vrma30qXPajmV0SkppUc/BpjmoGfAX9vrW0v9XHW2uustcustctyhcwiIlURBAUT3qrT6izf59dVza+ISC0qKfg1xqSIAt8fWWt/Hu9+yRgzO75/NrCtMqcoIjI6Uea3uq3O8plfo8yviEgtKqXbgwGuB9ZZa/+t4K5bgEvi7UuAm8t/eiIiYzCOrc5MflijzK+ISA0ppc/v6cC7gEeNMWvjfZ8GrgJ+Yoy5DHgBuKgiZygiMkrj0eqs//LGxjFYX8GviEitKKXbwx8BM8jdZ5b3dEREyqhYq7NKd3uwgCno9uCYfPszEREZf1reWESSq6DVWb4OocJlD2Fo850eQK3ORERKcc0117BkyRIWL17M1772tYqOpeBXRBKrsNVZrva30q3ObGDzJQ+gbg8iIsN57LHH+Pa3v80DDzzAww8/zK9+9Suefvrpio2n4FdEkquw1ZlbnVZnoe0b/EbdHio6pIjIhLZu3TpOPfVUGhsb8TyPV73qVdx0000VG6+UCW8iIhOStba31Vlcg1vpSNSGloLYVzW/IjKh/NPTm3lsf1dZn3NJcwP/ckTr4PcvWcIVV1zBzp07aWho4NZbb2XZsmVlPYdCCn5FJLmCgHwkmpvwVvFWZ/TN/KrmV0RkSIsWLeITn/gEZ511Fs3NzRx33HF4XuVCVAW/IpJYtmDCWzVbnTmuan5FZGIaKkNbSZdddhmXXXYZAJ/+9Kdpba3ceSj4FZHkCoLeVmcArlvxVmdhaHtLLNAKbyIipdi2bRszZ87khRde4Oc//zn33ntvxcZS8CsiyRWGGFMQ/DpOxSe82VAT3kRERuqCCy5g586dpFIpvvGNbzB16tSKjaXgV0QSy4Zhb5cHotKHirc6Cy2F8bYmvImIDO8Pf/hD1cZSqzMRSa7CVmcQBcIVLkEIbd9FLhyjml8RkVqi4FdEEquw1RnE7c4qnvnt1+3BdVTzKyJSQxT8ikhyFbY6g2jCW6UXuQiU+RURqWUKfkUkkay1YG2+1RnE7c4q3O3BaoU3EZGapuBXRJIpF+S6fbs9VGXCm1qdiYjULAW/IpJMQRTk5he3gCgQrkqrs97bCn5FRGqLgl8RSaT8YhZ9yh7cimd+w5C+Nb9O1AFCREQGt3LlSmbOnMmSJUvy+3bt2sVZZ53FEUccwVlnncXu3bvLMpaCXxFJplzm1+2X+a1wFjZaUrmg7MGo5ldEZDiXXnopq1ev7rPvqquu4swzz+Tpp5/mzDPP5KqrrirLWAp+RSSR8gtLFNQgGONUvNVZ/8yvyh5ERIa3fPlypk2b1mffzTffzCWXXALAJZdcwi9+8YuyjKUV3kQkmYpmfivf6qzY8sbF9ouI1KIrf/k4T2xtL+tzHjNnEp950+IRP+6ll15i9uzZAMyePZtt27aV5XyU+RWRRCpe81utVme9t3Pz7VT3KyJSG5T5FZFkGrTVWRUyv15BqUVB5hd3sEeJiNSG0WRoK+Xggw+mra2N2bNn09bWxsyZM8vyvMr8ikgi2SKtzozr5MshKiUMLI7bd8IboElvIiIj9OY3v5kbbrgBgBtuuIG3vOUtZXleBb8ikkz5sofCzK+LrXAUai2D1vyKiEhxK1as4LTTTmP9+vW0trZy/fXX88lPfpLbb7+dI444gttvv51PfvKTZRlLZQ8ikkxx8GvcgloDp/KLXIT9VnjLdX4IFfyKiAxq1apVRfffeeedZR9LmV8RSaR8bW9hq7MqLW/cv9VZbr+IiIw/Bb8ikkyDtDqr9vLG+W4PCn5FRGqCgl8RSaSirc6MqXirs3DQzG9FhxURkRIp+BWRZMoHvwULS7hudVqdFQt+1edXRKQmKPgVkUTKtzpz+y1yUeFWZzbs1+3BqOZXRKSWKPgVkWQq1urMrUars76ZX9X8iojUFgW/IpJMRVudmaq0OiustFC3BxGR4a1cuZKZM2eyZMmS/L6f/vSnLF68GMdxWLNmTdnGUvArIolkg2KtztyqtDor7DChCW8iIsO79NJLWb16dZ99S5Ys4ec//znLly8v61ha5EJEkiks1urMgQpnYAdkfo0mvImIDGf58uVs2LChz75FixZVZCwFvyKSSMVbnVV/wptWeBORCeW2T8KLj5b3OWcdC6+/qrzPOQbDlj0YY75jjNlmjHmsYN9njTFbjDFr48u5lT1NEZERqplWZ737RURk/JWS+f0ecC3w/X77/91a+9Wyn5GISBkUbXXmOhVf5GLQPr+q+RWRiaCGMrSVMmzm11r7e2BXFc5FRKR8cpnWwlZnxqn4hLdBV3hTza+ISE0YS7eHDxljHonLIqYOdpAx5nJjzBpjzJrt27ePYTgRkRHITXhz+k14q3Crsyjz23vbMar5FREZzooVKzjttNNYv349ra2tXH/99dx00020trZy77338oY3vIHXve51ZRlrtBPevgn8C2Dj66uBlcUOtNZeB1wHsGzZMn36i0hV5FudFU54q3CrM2st1vZb4c2NM7+BPv5ERAazatWqovvPO++8so81qsyvtfYla21go6WSvg2cXN7TEhEZo3FodWbzlRZFVnhT2YOISE0YVfBrjJldcPM84LHBjhURGQ/j0eos19Eh19u3cFvdHkREasOwZQ/GmFXAGcB0Y8xm4DPAGcaYpURlDxuAv67cKYqIjMI4tDrL1fU6rro9iIjUqmGDX2vtiiK7r6/AuYiIlM14tDormvl1lPkVEaklY+n2ICJSuwZtdVaF4Lew24NWeBMRqSkKfkUkmQZtdVbJml/iMYus8KYJbyIiNUHBr4gk0mCtzipZ9pCv+XWKTXir2LAiIhPeypUrmTlzJkuWLMnv+/jHP87RRx/Ny172Ms477zz27NlTlrEU/IpIMsXRZp9WZ061yh5U8ysiMhKXXnopq1ev7rPvrLPO4rHHHuORRx7hyCOP5Etf+lJZxlLwKyKJ1Jv57f2YM05lyx6KZX5V8ysiMrzly5czbdq0PvvOPvtsPC/qzXDqqaeyefPmsow12hXeRERqW9Ga38q2Ois24U01vyIykXz5gS/z5K4ny/qcR087mk+c/IkxPcd3vvMd3va2t5XlfJT5FZFEyrU6o7DVmWMq2+rMFil70CIXIiJj8oUvfAHP87j44ovL8nzK/IpIMuWCzT59xyqd+Y2HLNrnt2LDioiUzVgztOV2ww038Ktf/Yo777yzz2frWCj4FZFEsrmyB7d6rc7CYOAKb7lt1fyKiIzM6tWr+fKXv8zvfvc7Ghsby/a8KnsQkWTKTXhz+7U6s7Zi9bf5sgejsgcRkZFYsWIFp512GuvXr6e1tZXrr7+eD33oQ+zbt4+zzjqLpUuX8v73v78sYynzKyLJlGt1VjjhLVeLG4Z9guJyUbcHEZHRWbVq1YB9l112WUXGUuZXRBKpaKuzXMBbodKHIbs9KPgVEakJCn5FJJmKtTqLV3urWNlD0eWNNeFNRKSWKPgVkUTKd3Xo3+oMKpf5tUWWN84Fv+rzKyJSExT8ikgyFSl7yGd+K9TuLNftoTDz6xjV/IqI1BIFvyKSSLZI2UO+7VnFa34LM7997xMRkfGl4FdEkikXbBZ2dYgj0YplfvNlD4VDqtWZiEgtUfArIsmUy/wWrgiUy/xWKPjNZ3779/k1oJJfEZHBrVy5kpkzZ7JkyZL8vn/6p3/iZS97GUuXLuXss89m69atZRlLwa+IJJINBvbyzbU6sxUre8iN03cJTscY1fyKiAzh0ksvZfXq1X32ffzjH+eRRx5h7dq1vPGNb+Rzn/tcWcZS8CsiyRQGfducQW89QoXSsMUWuYCo9EFlDyIig1u+fDnTpk3rs2/SpEn57Y6Ojr7f5I2BVngTkUSyRVZxywfDlZ7wZvoHv6r5FZGJ4cUvfpGedU+W9TnrFh3NrE9/elSPveKKK/j+97/P5MmTueuuu8pyPsr8ikgyBWHfmWdQ8VZnxbo95G5rkQsRkZH7whe+wKZNm7j44ou59tpry/KcyvyKSCLZImUPpsIT3sIiyxtDVAahml8RmQhGm6GttHe84x284Q1v4MorrxzzcynzKyLJFNoBZQ+5THDFJrwVWeENwHFV8ysiMlJPP/10fvuWW27h6KOPLsvzKvMrIskUBgMnRzgVbnVWZIW33G1lfkVEBrdixQruvvtuduzYQWtrK1deeSW33nor69evx3EcFixYwH/913+VZSwFvyKSSEO1Oqtc2UN0PSDzq+BXRGRIq1atGrDvsssuq8hYKnsQkWQKw4Gtziq8wluu7KF/5tdxTT4rLCIi40vBr4gkkg2DAd0e8hPeqt7qTJlfEZFaoeBXRJIpCHuXM87JtToLKt3qrN+wjiFU5ldEpCYo+BWRRIpanfWv+c2t8FbZVmduv6BbK7yJiNQOBb8ikkyhLZL5rWyrs1x217gDa35V9iAiUhsU/IpIMoUBZmD9QXxfZTO/xbo9KPMrIlIbFPyKSCKNR6szO0jwqwlvIiJDW7lyJTNnzmTJkiUD7vvqV7+KMYYdO3aUZSwFvyKSSDbwe4PdnFyrswpNeBuy7EET3kREBnXppZeyevXqAfs3bdrE7bffzvz588s21rDBrzHmO8aYbcaYxwr2TTPG3G6MeTq+nlq2MxIRKYcgBG+QCW9hhWp+863O+u5X2YOIyNCWL1/OtGnTBuz/yEc+wle+8pWBK3aOQSkrvH0PuBb4fsG+TwJ3WmuvMsZ8Mr79ibKdlYjIGEWZ334fcblWZxUKRG1gcRxTtM9vkKlMtrmWBGGA26/DhohMLH/4yVPs2LS/rM85fV4zr3zrkSN+3C233MLcuXM57rjjyno+w2Z+rbW/B3b12/0W4IZ4+wbgr8p6ViIiY+UHA8oe8iuvVTDz27/kAQ6Mbg+PbH+E5T9ezuW/uZxMkBnv0xGRCa6zs5MvfOELfO5znyv7c5eS+S3mYGttG4C1ts0YM7OM5yQiMmY2CAZMeMvdrmSrs/6T3SD5ZQ/WWj5/3+dpz7Rzb9u9/O9T/8s7Fr1jvE9LREZhNBnaSnj22Wd5/vnn81nfzZs3c8IJJ/DAAw8wa9asMT13xSe8GWMuN8asMcas2b59e6WHExGJBAMzv9VodeYUyfyahK/wtm7XOtbtWscVp1zBsdOP5RfP/GK8T0lEJrhjjz2Wbdu2sWHDBjZs2EBraysPPvjgmANfGH3w+5IxZjZAfL1tsAOttddZa5dZa5fNmDFjlMOJiIyMDQJM/wlvuUUuKtXqLCge/Ca97OGPW/4IwNkLz+bM+Weybtc6Xup4aZzPSkQmkhUrVnDaaaexfv16Wltbuf766ys21miD31uAS+LtS4Cby3M6IiLlEZU99KvsymWCK1X2ENreuuICSV/e+L62+zh62tFMq5/GK+a+AoAHXnxgnM9KRCaSVatW0dbWRjabZfPmzVx22WV97t+wYQPTp08vy1iltDpbBdwLHGWM2WyMuQy4CjjLGPM0cFZ8W0Skdvh+PtObY7woGK5YzW84eM1vWKHewuPND30e3f4oJx58IgCHTzmcRq+RR7Y/Ms5nJiJS3LAT3qy1Kwa568wyn4uISNnYIACv70ecqXDmd9CyhwSv8Pb83ufpDrpZfNBiAFzHZcn0JTyyQ8GviNQmrfAmIskUFpnwFpdBWL/KZQ+uwSYz8csTO58A4JiDjsnvWzRtEc/sfoagQi3lRETGQsGviCSS9YtMePNyrc78iow5VKuzpGZ+n979NGknzcJJC/P7Dp1yKJkww9b9W8fvxEREBqHgV0QSqdiEt4qXPQzS6sxxDDahrc42tm9k/qT5fVZ2O3TyoQA8t/e58TotEZFBKfgVkWQqMuEtVwM8HmUPSc38bmjf0CfrC3DI5EOAqB5YRKTWKPgVkUSyYQj9yx7ymV+VPZRDNsyyed9mFk5e2Gf/5LrJTKufpsyviJRs5cqVzJw5kyVLluT3ffazn2Xu3LksXbqUpUuXcuutt5ZlLAW/IpJINvAxg5Q9VCzzG4SDLnKRxLKHLfu24Ft/QOYXotIHBb8iUqpLL72U1atXD9j/kY98hLVr17J27VrOPffcsoyl4FdEkqnIhLfesofKZH7tEItcJDHzu6F9A8CAzC9Ewe/ze5/H2uT9u0Wk/JYvX860adOqMtawfX5FRCaioSa8VazbQ2hx3IE5BSde4c1aizEDg+OJasPeDQDFM79TDqU9087O7p1MbyjPqkwiUnl3fe86tm0s77c2MxccyqsvvXxUj7322mv5/ve/z7Jly7j66quZOnXqmM9HmV8RSaZiE95yNb8VK3sYpNtDvC9pSxxvaN/A1LqpTK6bPOC+eS3zANiyf0u1T0tEEuIDH/gAzz77LGvXrmX27Nl87GMfK8vzKvMrIolUdMKbMeB5FVve2A6yvHGuFCLKDFdk6HGxad8m5k+aX/S+OU1zgKgu+LgZx1XztERkDEaboa2Egw8+OL/9vve9jze+8Y1leV5lfkUkkWwQDJjwBnHpQwXLHgar+YUoM5wkbR1t+SC3vznN0f6tHVroQkRGp62tLb9900039ekEMRbK/IpIMvn+wAlvRMFv5bo9DL7IBUCS5n6FNuTFjhd57YLXFr2/MdXItPppbN63ucpnJiIT0YoVK7j77rvZsWMHra2tXHnlldx9992sXbsWYwwLFy7kW9/6VlnGUvArIoljwzCKNIvVGIxD2UO+5jdBmd9d3bvIhllmN80e9Ji5zXO1xLGIlGTVqlUD9l122WUVGUtlDyKSPHErs8Eyv5Vc5KJY2YNTUPObFG37o68jhwp+5zTP0YQ3Eak5Cn5FJHFsGEYbbrHMbwXLHsLiZQ9JrPnN1fIOm/nt2Epow2qdlojIsBT8ikji5ILb4hPevIr1+R2u7CEMkxMEvtjxIgCzm4cOfv3QZ1vntmqdlojIsBT8ikjyBMOUPVQo8xsEFjPUhLcklT10tNGUaqIl1TLoMXOb5wKo7ldkApjIqzGO9NwV/IpI4uQntBWd8OZWbsJbMEifXzd5ZQ9t+9uY3TR7yBXrcu3OVPcrUtvq6+vZuXPnhAyArbXs3LmT+vr6kh+jbg8ikjxxcFs881u5sofBan6deKW5RE1462hjVtOsIY/J3Z8rkRCR2tTa2srmzZvZvn37eJ/KqNTX19Pa2lry8Qp+RSRx8pndIhPeKln2EA5W85vACW8vdrzIsdOPHfKYBq+BafXTaOtoG/I4ERlfqVSKQw45ZLxPo2pU9iAiiTPUhLeK9vkdbJELL1nBb2e2k909u4ec7JYzq2mWgl8RqSkKfkUkeYaZ8FbJsoeifX5zNb9+Mro9vNgZlTEMV/YAUSu0XE9gEZFaoOBXRBJnqAlvlSp7sNYO2urMdeOa34Rkfl/cH7c5G6LHb87sptm0dbRNyIk0IpJMCn5FJHmGmPBWqbKHXBuzomUP8b4gSEbmN1fGUErwO6tpFp1+J+2Z9kqflohISRT8ikjiDD/hrfxlD7msbtGyBy9Zmd+2jjYc4zCzceawx+YCZHV8EJFaoeBXRBJnyBXeKtTnN4gDW9cb+LHq5ia8+ckJfmc2zsRzhm8YlAt+NelNRGqFgl8RSZ7chDe3yEecW5myh9xktmLBb67Pb5LKHkopeYDe5Y8V/IpIrVDwKyKJ01v2UCTzW6Gyh8AfouY3Ya3O2vYPv8BFzrT6aaSdtIJfEakZCn5FJHmGnPBWmbKHMBgi85ub8JaAVmehDXmx88WSM7+OcaJev2p3JiI1QsGviCROLrg1RSe8ediKZH6jwDaX5S3kJmjC286unfihz5ymOSU/JtfuTESkFij4FZHEyU14G49uD26ROuPePr8TP/O7tWMrQEmru+VolTcRqSUKfkUkcWw2A4BJpQfeWaluD/nM71BlDxM/85sLYkut+YUoUN7euZ1smK3UaYmIlEzBr4gkjs1GQZZJpwbcZyrV7SGf+R1qwtvEz/yOZHW3nNlNs7FYtnVuq9RpiYiUTMGviCROPvhNFQl+Pa9C3R6Gyvwmp+a3raONllQLLemWkh+T7/WrSW8iUgOG71A+BGPMBmAfEAC+tXZZOU5KRGQshgp+K9btwR8i8+sYjElGt4e2jjZmNZde8gBa6EJEasuYgt/Yq621O8rwPCIiZTFk5rdS3R6CwTO/uf1JyfyOpOQBeuuDFfyKSC1Q2YOIJM5wZQ+VCH7zmd8irc4gmvSWhOWNt+zfMqI2ZwD1Xj3T6qcp+BWRmjDW4NcCvzHG/MUYc3k5TkhEZMyGCn7TaWwmU/Yh85nfYksqE7U7m+gT3vZl9rEvs485zSMLfkHtzkSkdoy17OF0a+1WY8xM4HZjzJPW2t8XHhAHxZcDzJ8/f4zDiYgMb8jMbzoNvo8NQ4xTvi+/wtyEtyI1v7n9wQQve9i6P+rxO5rgd3bTbDbs3VDmMxIRGbkxffJba7fG19uAm4CTixxznbV2mbV22YwZM8YynIhISYYMfuN9uWPKJRfYFlveGKJ2Z+EEn/CWy9yOtOwBeld5s3Zi/wEgIhPfqINfY0yTMaYltw2cDTxWrhMTERktm82CMcVXeEtHC1+Uu/RhuMyv6zoTPvO7Zf8WYGSru+XMappFp99Je6a93KclIjIiYyl7OBi4yRiTe57/sdauLstZiYiMgc1mMakU8edTHxXL/PrDZH5dM+Frftv2t1Hn1nFQ/UEjfmyuVOLFjheZXDe53KcmIlKyUQe/1trngOPKeC4iImVhM9niPX7pXfWt3Jnf3glvg9T8JqDV2daOrcxuml30j4rh5Nqjbd2/laOmHVXuUxMRKZlanYlI4uQyv8Xkyx7KnPkNh8n8uq7JZ4cnqq37tzK3ee6oHqtevyJSKxT8ikjiDBn8psYp85uAVmdb928dVb0vwLT6aaSdNC92vFjmsxIRGZlyrPAmIlJTSsr8ln3Cm8VxzaAlAa5nJnTZQ2e2k909u0ed+XWMM+F6/Qa+z/MPrWHjo2tp3/4SxnGZOnsOhx6/jNZFS8raKk9EqkfBr4gkTkmZ37K3OgsHXdoYooxwpjso65jVlAtaR7q0caFcu7NaZ61l3R/u4g83fp/9O3eQqqtnyuw52CBgw8N/Yc0vf86M+QtZ/q7LWPiy48f7dEVkhBT8ikji2Gw2P7GtP6eCmV93kJIHiGqBA7+8AXc15Ra4GG3mF6K633vb7i3XKVVET2cnq//z33jmz/cx+/CjeO1lf8PC407A9aJfl9nubp66/x7u+/mN/OwL/8TS172BM979vvz9/YWhxQ8t6SH+MBKR6lLwKyKJYzMZqPKEt+Eyv27KJchO3JrfXPA7lszvnOY5bO/cTibIkHbT5Tq1smnfvo2ffekz7G7bwhnvfi8nvP7NA0obUvX1LH7VmRx12iv5443f5y+//gW727bypo98irrGRqy1/OnZnfzioS3c//wuNu/uJLTQUu/xstbJnH3MLP5q6VwmNxZ/f4pI5Sn4FZHEGY8Jb6EfDpn59dIO2czELXvY2rEVz/GY0Tj6lTrntczDYtm8fzOHTj60jGc3du07tvGTz32K7o79XHjF55m/5GVDHu+l05zx7vdyUOt8bv/2tfzsC//EIe/+GP965/M8+MIeWuo9Tj9sOm8+bg5pz+Gl9m7WbNjNZ255nKt/s573n3EYK08/hPrUwIVYRKSyFPyKSOKUMuEtLHe3B98O2uYMwEs5Ezrzu2nfJuY2z8Uxo//6fv6k+QC80P5CTQW/nXv38NPPXUH3/v1c+I+fZ9ZhR5T82GNfczZuUzNXfO9uHvruQ8xsSfPF847l/BPmFg1sH9uyl3+//Sm+sno9t6zdytdXHM+RB7eU858jIsNQEZKIJE4pmV/KXPbgZwK89OBZPC/t4mcmbvC7sX0jCyYtGNNzLGiJHv9C+wvlOKWy8LNZbr76i+zfvYvzP3XliAJfgO37evjUGstDk4/j2PbH+GTTw6w4ed6gGd0lcydz/aUn8d33nMSO/T286T/+yK8fqf1JgCJJouBXRBJnPDK/UfA7dObXn6CZ39CGvND+wpiD38l1k2lJt/DCvtoIfq213PHf32Dr+ic4528+wpwjjx7R45/bvp/zv3kP619s55sXn8AVr13I8/fezV9+ddOwj331UTO57cPLOXbuZD606kH++w/PjfafISIjpLIHEUmcKPgtPqGqUn1+/Ww4TObXwYaWIAhx3YmVd9jWuY3uoDufuR0tYwzzW+bXTOb3odtu4fG77+C0C1dw1GmvGNFjn9/Rwduuu48gtNx4+WksnTcFu+StbNvwLL//0feYvuCQYdugzWip44fvPYWP/mQtn//1OrKB5QNnHDaWf5KIlGBifQKLiJRgPPr8+pmQ1BCZXzf+GjyYgKUPG9s3ArBg8tiCXyAKfmsg8/vSc8/wux9+l8OWncppF6wY0WM37erkHd+OAt8fX34qS+dNAaLg/py/+QgHzZvPr6/5Cvt27Rj2uepTLv+x4gTesnQOX179pDLAIlWg4FdEEqe0Fd6qXPObij5uJ2LpQz74HWPmF6JJb20dbWSD8et5nOnu4tdf/wqNU6bwug98eEQrtW1r72bFt++jMxPww8tO4Yh+k9XS9Q286SOfIshmue3afyMMh+/w4TqGqy86jnOPncXnf72OVQ+M/x8HIkmm4FdEEsdmMoMuclGpzG92uJrf+D5/ArY7e6H9BercOg5uOnjMzzV/0nxCG7J5/+YynNno/Pa732L3i22c+6GP0dBceqeFrkzAe7+/hp37M3x/5ckcM2dS0eOmzZnLa1a+n02PP8Kfb/5ZSc/tuQ7XvP14Xn3UDP7xF49x1/ptJZ+XiIyMgl8RSZywqwunobHofRWr+c0MU/Mblz1MxI4PG9s3Mq9l3pjanOUcMukQAJ7bMz5f76+753c8fvcdnHr+25h3zLElPy4MLR/9yVoe3bKXr684nuPiUofBLH7VmRz18uXc85MfsvWpJ0saI+U6XPuOEzh6Vgsf/NGDPLZlb8nnJyKlU/ArIokTdnbiNDQUvc84DnheBYLfocse3HzZw8TL/D7f/jyHTD6kLM912JRoQtfTe54uy/ONxJ6XXuSOb3+DOUcuGnGd77/+Zj23PfYiV5y7iLOOGT4DbozhrPd9kJaDZvDrr/8rPZ0dJY3TVOfxnUtPYnJDistu+DNb93SN6DxFZHgKfkUkUWwmA76P01g8+AVw0mlsT0/5xrQ2yvymBv9ITcWB8USr+e3MdvJC+wscMXVk/W8H05hqpLW5lWf2PFOW5ytV4Pvc+vV/xRjDuX/7Dzhu6Sur/WTNJr5597O845T5XPaK0v8IqGts4g1/9w/s27md27/9Day1JT3u4En1fPc9J9HRE7Dye39mf49f8pgiMjwFvyKSKGFXlClzGouXPQCYpkbCzs6yjZlbuW2oml83vm+idXt4Zs8zWCxHTj2ybM95+NTDeXp3dTO/f/rpj2h7Zj1nXf4hJs8svXb53md3csVNj/LKI6Zz5ZsXY8zgS1gXM+fIRbz8ootZ/6ff8/jv7iz5cUfPmsQ3Lj6Bp7ft5+9WPUQQlhY4i8jwFPyKSKLkgl8zSNkDRIFxOYNfPx/8ltLtYWKVPTy1+ymAsga/R0w5go3tG8kE5S09GczGR9fywM3/y5JXn81Rp72y5Mc9t30/7//hX1hwUBPXvuMEUqPsz3zyX13IvGOO5bff+S92bd1S8uNedeQMPvvmxfz2yW18/tdPjGpsERlIwa+IJErYGWd+B5nwBuA0NhF2lFaDWYpcB4dUKRPeJljZw1O7n6LRa2Ru89yyPecRU48gsAHP732+bM85mM72vdz2jX9j2uy5vObSy0t+3O6ODJfdsAbXMXznkqgGd7Qcx+X1H/oYbirFrf/xrwR+6Z1G3nXqAt5z+kK+e88GfnDvhlGfg4j0UvArIokSdkUZ3SFrfstc9pDr4JDEVmdP7X6KI6YeUZZODzmHTzkcqPykN2st//fNr9G9fx9v+PD/I1VfX9LjurMBl/9gDVv2dHHdu05k/kGD/yFVqpaDpnP2+/+Ol557hj/e+IMRPfYf33AMrzl6Jp/95RP87qntYz4XkQOdgl8RSRQbB7WDdXuAuOyhjJnfbBzQDlX2kKqL7sv2TJzg11rLU7ufKmvJA8DCyQupd+t5fMfjZX3e/h667Raee/DPvOqdK5m58NCSHhOGlo//7yP8ecNurr7oOJYtnFa28znipNM47qzXs+aXP2fDww+W/DjXMXx9xfEcMbOZD/3oQda/uK9s5yRyIPLG+wRERMqplAlvTlMT2U3lW2Qhn/kdottDujH6uO3pnDgz9ze2b2RfZh+LDlpU1udNOSmOOegYHt7+cPEDMp3QubPgsguyneD3QNADfjdYwPXA8cBJQaoe6idD/RRomMK2l/bwxx9dz2HLTmHp695Y8rldfft6fvnwVv7fOUfxpuPmlOXfW+hV734vm9c9zq3XXs07v/Q1Jk2fUdLjmuMWaG/5xj2s/N6f+cUHT2dGS13Zz0/kQKDgV0QSJVfza4as+S1v2UOmKwpo0w2Df6S6roNX506o4PehbQ8BcOLME8v+3KdMOZK/PPoDsg98i9SezbBnI+x5AXZvhK5dY37+mcDfHg6wDvPfd8GkuTBlPkw/EmYcFV039s3q3vjAC3zjrmd5+0nz+MCrDhvzORSTStfxpo9+iv+54qP88t++yNs++2W8eOGV4cyZ0sD1lyzjrd+6l8t/sIZV7zuV+lTpLdtEJKLgV0QSpTfzO1TNb3knvHV3RBOY6puGnhRV3+jR01neZZUr6cFtDzKlbsrYF7jYswleuA/a1sK2J2DbOv5mX1t039b/B24aJs+DqQtg9nHRdtMMaDyo95JuBK8evDpw68A4EGYh9CHIRtngrj0EHTu453vX0rPzBV5+9hk0sR/at8D2J+Hp30TH5TTNgJmLYPZSHvIX8q0/wCsPX8y//NWSEbc0G4mD5s7jnA9+lFu++gXuuP4/ed37P1zyeC9rncLX3raUD/zoQf76B3/hunefSJ2nAFhkJBT8ikii5Ce8ldDqzFpbliCne39pwW9dozfhMr9LZy4d2WsUhrBjPWz8UxTwvnAv7N0U3efVR1nXQ8+gY+oCPrbuel5z4gd567IPgzOKKShu319htmU2t//XNTy+vpM3/N2XaTr9VQPPbe8LsP2pKBjesR5eepzwvm9yfJjlrjqwOyZjfngctJ4M80+DeSdF5RRldsRJp3Hq+W/jvp//mIPmzuOkN19Q8mPPWTKbq84/lk/87FH+9n8e4hsXj74Nm8iBSMGviCSKzWV+hwp+m5ogDLHd3UP2Ay5Vd2cWTG9d72DqGlMTJvjd0bWDje0bueCIYYIyPwNtD8MLf4KN98Km+6Brd3Rf88FRAPnyv4X5p8LBS8CJspRNwAu7/sDv9z3LW0cT+BZxz49/wON338GpF7ydo/sHvhAF2FMXRpcjzwbgD09v5wM33McZU3fylZcHNO54FLY+CH/8d7BfBUx03vNP7b1Mbi3L+b78oovZ1baV3//ouzRNncYxr3x1yY9920nz6coEfPaXT/DRnzzM1962FNepXLZaJEkU/IpIogT79oHjDLvIBUDY2TlkkFyqnv1Z6ho9nGGCj7pGj/Yd3UMeUyvu3XovAMsOXtb3jp59sOmB3qzu5jXgR39wcNDhcPQbYP7LYcFpMPUQGCJr/Iq5r+Cmp2+iy++iwRvb/8NDq3/J/Tf9hGPPfB0vv+jikh6z+rE2/m7VWg6dMZkr33s2jc0FE8h69sOWv/T+Ox9eBX/+dnTf5Hmw4OVRYL/gdJh+xJD/zsEYx+H1H/woXXv38H/f/BqNLZNYuLT0+upLTz+Ebj/kqtueJOMHXPP241UDLFICBb8iUnNCa+kOLd1hSHcQ0hWG0e14uyuIbgfW4ltLYCGwlgDLvBe2MHXKFK7bsgPfRs/lGIMDuAYcY5jZE3IE8LNnN+H3RHFLyhjqHId6J7quc3pv1xfcbnSjfYWlAN0dWeobh18EoW4C1fzesfEOZjbOZHH9DHji5iir+8K98OIjYMOo5nbWy2DZe+KM6GnQPHNEY5wx7wxWPbmK+9vu54x5Z4z6XNf+36/57Xe/xWHLTuG1l/1NSWUaP/7zC3zq54+ydN4UvnvpyUzu//9X1wyHviq6AAQ+bHu893V49i545MfRfY3To2B/wenR6zDr2HyGezheKsVbPv6P/Pizn+QXX/08b/rIpzjsxJNL/re//1WHkXYdPverJ3j3dx7g2+9eNqYFOUQOBAp+RWpANsjSke2g0+/MX/f4PfQEPWSCDD1BT59Lbl/uOhtmCWxAEAaENsS3PqENCW1IEAbRfTa6z1qLYxyMMTjGwcEBAw7OgP2u45J206ScFCknRdpNRxcnnd+f3+emaXAbaEw14rn19Ng03TZNp03RRZqu0GWfH9DuB+wL4ms/zN+O7gvpCAK6Qzvq1/Lzm7Yys7GFzzyzddBjTuoM+Apw3SNP8fgo5r15Bppdl2bPodl1yc7ySR9Ux+8e2xDvi/ZPSblM8Tymplympjx2NLvs9AOyoSVVi19RWwu7nqPn+d/zmod+zj8HLs7VR0X3eQ3Qugxe+Q9RoNd6EtS1jGm4kw4+iaZUE3dvuntUwa+1ljW//Dm//9F3OfTEk3njhz+B4w4ddGaDkC/euo7v3rOB5UfO4L/eeQKN6RJ+FbpeNBlv9nFw6vvzrxUb74nqmzf+Cdb9Mjq2bhLMOznKDi84HeYcH03UG0RdYxMX/dMX+NkX/5lbrv4C5/7txznqtFeU/DqsfMUhHNSc5h9++jAXfPNP/Nc7T+Twmc0lP17kQGOsHf0vmZFatmyZXbNmTdXGE6mGbJBlb2Yv7T3t7M3sZW9PdGnPtLO3Zy/7Mvt6g9psdN3hF2xnO8iGo8sG1rv1pN00nuPhGQ/HcXCNi2tcHBNvO31vYwBLFBwTBcOhDbHYfHAcEgXOfuiTDbL0hBkyQXQJ7OhqVi0u1tRhnXqMU4/r1OO6jaS9Zuq8JhpSzTSmWmhKNdOUaqY53UJLfJmcamFy/SSmpJtpclPUu1E2NmUMrjG4Blxj8Ixh38XvwJk8mRnf+hYuYIyJ/21RFjgAMuufYveFF9D0lX/FO/tsQiAbWnpsSHdg6QlDeuLMc0/Ye7srDOkMQvb7AfuDkH1BQIcf8swzu8mmHbzpdewvuH+oT9cW12FKymOqFwXJU1MeUzyXg9Ie01Me09Mppqe8/O2pKRen3B0I/EyUyd10f/T1/qb7Yf9LAOxxHOy8U5h65OujMobZx4FXWkuukfj47z7OfW33ccdFd1Dnlt63NvB97vzON3n0zv/jyNNeybkf+iiuN3TG86X2bv7+xrXc+9xOVp5+CJ8+92i8ck4U27sZNt6L3XgP/vP3kt32LJnQJWMayB60GH/6MQTTjyKYchgBLkE2S+Bno+tslp6uTh6/+w727dzB3KOOYfqChYS+TxgEhGGIDUPCMIT42tpoX27/no4ennqxnTAMOXR6I9Oa6jAm/pai33Xx/Q6O6+B6KRzXxfU8HC+F67o4nhfdduPr+HZun5dK4dXV4aXrSKXr8NLp3tt18e10Ha6nnJtUjzHmL9baZf33610oQpRB6vQ784Fr/2C2vac9H8zm98W3u3L1jkUYDM2pZhpTjTSlmmhKNdGYamRa/bT8dmOqkSavqc/9DV4DDV4DaTdNnVuXvxTeTjnRL/rAD/EzuUuAn42ug2xIEISEge29hCGBb+nKBuzL+OzPBuzPBHRkAzoyAV3ZgC4/oDsb0uMH9PghGT/EBgAWY4kuhBhrcQlJA3UGUsaSwuIR4toQlxDHhhgbHWttAISE1icgykT71idrs/hhlqyNLtCJNR1kaWOXsewELBZMFKC7rovnuaQ8j3QqRdpLUZ+qoy5dR326jnmZVsK6GTz1mwdoqmukqa6R5romUqkUjmtwXAM9LeyddAjepr007crgpRxcz8VLebh1Dq5ncEYQFH3vh1uYt2gaZ76hdzGI0Fr2+QF7/IDd2YDdWZ8nntzJn+/dyuGvm0dPvRPvD9jj+2zp7mK377M7GxQNmh1gWspjej449jgodzved1AuaE57tLjOwK//92+HzQ9EQe6mB2DLg9GiEQBTFsAhy2H+aVy59Q7+0P0Sqy/6VbSIRAVdeOSFrN6wmlufu5XzjjivpMfs3fYSt/7HV9n61DpOOe+tnP7Wd2KGmDRnreWmh7bw2Vsep8cPufqi47jgxL6T1sIgINPdRaari2x83dPVSbarK97fSSa/3XtM//tyx4fBDKBgAYtnANbFl2EYw5b1T9D27FPUN7XgpqI/bI3jYBw3+nYmf9uJto1Dk+uwZE4Lz2zr4KntnczoDlg4rRHPARsGWAvWhmAtNrqBLbjkguow8Al8n9CProM4AA/8LDYMS/o/GvSf5jhxMJy7pEnV1ZNuaCBVX0+6viHebshvp+sbSMXX6fp60g2N8f3x4+rqh/z/F+lPmV9JFD/080FpPliNtwfLzObu84fIaKadNJPrJjO5bjKT0pP6XE+um8zk9GQm1U2KrlOTaHJbaDLN1NNAmAU/G/QGp5kwClAH7AvwewKyPdG13xMUDWj9bIjvh1HQ65f/5zcErAPWEGeDoppYx0YBmEO07ca3c+FVYZhlsGB75wCZgnttHNrlPnpsfMlth8SZZ2OJwl2b32/jgywGE5+kQ6Um+FhcJ8oou07UKMB1CrZdg+cYHBfad/k0tXi0TPHwPAfPM6RSDm7aJZUyeGkXN2Xo6Qp5/P5dHHv6dA4+pJlUvUuq3iPVmMJt8HBSLoHrsgfLDmvZlQ3Z4YfsyPrsyAbszPrsyMSXbJadWZ92v3gwMjezi5d3P8sJHU+zeN9THLb3SQ7qjHrrBk6K9hlL6J57Mmb+KTTOP4WWqXMxxrB+13ou/OWFfHDpB3n/ce+v0Gtb8Cpby/m3nI9rXH76pp8OWa8bhgGP/fZ2fvfD72Ct5TXvuZxDli4bEJjmg9CuTrZs38O9T25h2852Dm6AxTPr8YJMn0A109WFn+kp6XyN48QBWWOfwKyuobE3aGsoCNwaGqNj6utxyeLufBp3+2N4Lz2Mu+0xXNuNayzu5Nm4c47FbV0a9R1+ZDN3/+wXNE2Zylnv+xCHnnBSya9pxg/5xl3P8J93P0NLfYpPnHMUF5zQWpYstw1DgiAg9LO9QXE2i5/N4md68DM9ZHt68DOZ+HYmvt2D39ODn41vx/uy8fGFf0xku7tH9H8C5APo3P9BXUNjPmCua2zM70s3FPx/1TeQboyvGxrjS8OwpTMycQyW+VXwKzUntCH7s/vzwWrhdS5Q7RO4Fmx3ZAcp4LQGL0wxxZ3KFHcqk72pTHIm0+JMptmZRCPN1IeN1IUN1IX1pPw6XD+N53u4fgqbjZawDbLxxc9dbHQdWMIgvh5DYsQlmpSFARxD6EDgGnwXMp4h4xq6U9DlGTpTho60YX/KkEk5ZF3wXUPWNWQ98B2DR0hTGMQXnyabpdlmaA4ztNhuWmwXk2wXk8NOJrGfetONcX2sk8XiY40PxseaIL72owysibK4mABLiDFRyIrp3ReliEOsCSGXuTVxGGuIjsVG+wzx/lxqOQ5089v5lDMU5kVNVM6AdQhDl+x+l3lX1vHC62HjyS5doUu379EduHQFLj2BRyZw6Qk9LrrJ4cVpaX59UhrHpnBDL76kcG3vdtp61FmPtE2Rti6pMIVnPbyC45wwhQlSmDANQXwZBeP24LhZjJvFcTO9207vbcf1MU4WN96P8XFtgBv2UO930xzs46DMLiYFe/FMDymT4aX6STzdPJdHJx3Gn6a+jEcmHUmP07fEwLM+k8IO6oN26oK9LLDdTAv2MyXoYGrQyRS/gylBJ9P8TqYG3bQEGRyc+I8QFxNfsH335W6b0CUIHYKsxQ8Mvm/I+hY/sGzet42ndj3PIS2HMCU9lWwQRJcwwA8CsoFPp5+hw88SWpur3ClZ1nikjEOzsaQcB884pIyD55je7fy1ie9zSGH67PeMwaW3TCB3MU78Q+s48T4wJt52opr6KDOZ+0syfhwhKfMiKeKLfQnP7u49b5pp66znpa56AmcGs2YsprllIdZpiB7vOuC4A65xowzxMxmXf9lYxyMdDgvqLe9fYDj7YIdUysO4btQVZcjrgc9tXAdcN/r39L/2ouc1nhdtj7FMJwwCsj1RIJzPtnf3244D5Wx37g+g7t4/hro6e/8g6uokyJZWXual6wr+gGnMB9XpogHzYPdF+1TmMb4U/EpeVOMJQWgJQosfhvnt6PbAbT8MCUPyx/qhJexzf3wdhHQFXXRmO+j0O+jy4+ugg26/k+6gI3/JZrvx/Qy+30Pg9xD6GULfB5vFCVN48cXtt50O6kmHjaSDeuqCBlJhHamgLgpKgnQUtFgXJ3QxYfyLmNF+CFscE+I4ASZ37QQYE0Tbro9xfBzHB8fHcbPgZMHNkvUsGQ96PEuP59CdMnSlXLo8l04vRUcqRUcqzX63jvZ0Pe1uM/udJnpMfdEzMTakmf200E4L7UyKr1vYRwt7aWFfvG9v/r40Y+8sEH076mCtQxhGr6W10aVwu+/tKCtrBzl2wH3Wyd8eeJ/pc1/0vxL/f+ZvAxgmbehk0aotPHnRXPYc1tTvmNxjou3FN27EzYQ8/K5D6CGkB0uXDeg2Id0EdJuAbhvSY3x6TEgPQX5/D9Exvhnk89OCG6ZoDOpoCutpDOtoDOuoD+uoD9PUh3XUhen8+zYVpknF7283TMfXHiZMQZiCwMUJPAg8CD1smCII04R2NEF2CK4Prk/ohoRuiO9afA8yriHjufS4Hl2uS5eBrIGQDNZmsWTAZoEMTtBDOuwkbTtJBZ2kgy5SQXd08Xvw/Cxu1sfJBpishayFoLSfQ+NYnFQYXbwQJ2Wja8/ieiGuG+K4Fte1OK7FcUMcB3w8nsss4C+dS1ifPRTrGl7R/BBvaP4DzbY7+tssNBCY/LYJ4r/ZQgcT2Ghfbn9gcKzByR/n4AREt32DCSxO7jG+gQCIH2d8G5ULWTBhrsQgjL/eCHM/WPnyg9w+4/rUT+qhfko3dZN7qGvJkJrk4zq977XuTo/s3uiPvex+j2yHSya+DrN9s7sWuG/WYr6/6Bw2TJ7N1O52ztl4P6/e9CDz9m8fxftnBHJBdCoVBcWuGwXFcZCM52JcLw6W4+3C4Dl3jBc/3nMhPj7aLvJ4Lx7D7XdMyiMw4IcW34b4NiQTBPhhQMbP4gch2SBLNpsl40fX2WyGbCZDJs5QZ3p6yPR0R5nrbKakl8BNpXqD5D4B8xD7CoLpXNY63dCIl1IXj5GqSPBrjDkHuIYoYfXf1tqrhjp+vILfwYK9wQK44QLC3tth0UCx/zFByIBj+weWwx0TBLlzjiY+BGGcaQxDwngcG+ZuR5OVwtBiw4DQ2nhChMUJQ5zQkjIhddaScjJ4JiCFj2MCPBPg4uM4Aa6JAz7CqMbTWIy1vV9zWwAHYw0mjLI8jnXzgasbenhhumgA69kx/BCbAONmMU6UFYsuGRw3k8+YGTeD40X7HK8H42ZwUz1973ezmPgY3Cy+awlc8F3odj26PY8ep44u6ummgS4a6KaB7vh2N/X5fYXXHTTRZZqG/Cd4Nkuj7cxfGmwXzWEHTWEHzeF+msNOmoJOmm0HzUEnTUEXjbYHxxrAieoSojxx/H/gRq2nrIcx0f7cdbTPwVgPgwfGweBhcKPsnPEw1sMSH2Pjx9sUJp/Zi+p2CS1O7pe1DTFh734TT8Sx8XXudu8lgCDeDnLHBPnbJgyw8XX0AxFgAh8bhpggxAZB9BxhiM3dHwbYguMb9+yhub2d7TNm0Cc3WORjrrGjg8auLnZNnYodol7QDPEZ6TuWpw59JVtnLGD+hh/RnQ7oqLd01lk66qCzjvztzrSlq86yr8Gjx8sQOMGQ7xEAx1oaQ0uzDWkOQ5oCS3MQ0hgYvEwKk6nHZpuhp5kg04TNNkY/f4GDG7q4gYNrHZzQRGUqNnpOk/v/swGGAGwA1gf8aLlgm422S+aCSYFJYUhH1yZN6KSwTprQSRO4aQK3jsCLtkM3Reh5WDdF4LpYz4WUA57BuBbHyeKaDI6TwTM9eE43KaeblNNFyunGc7twTQ+7expp65rGxs5ZPLt/Hj3WpaV+Ly+ffR+nzb6fSXUdOCbEONE3FMZYHGdsdasjFYbxH3jxH4a24DY29wdlXGtkcz/fudojBxMaGnqyNHR009DZTUtPN5OzPTQHPaTDvu+jjJuiO9VAxmugJ9VAxmsk4zXS4zWxrnsud+w9knvbW9lPA7Pruzl+8l6OaurkyOYMB6UthlRcSuThWA8HL87cE/0hkN8O48+DgusgwARB9LMfhBg/G31G+AEEPsaP7s9t2yDA+j4EPtYPsIEPfm47iLf9aDt3jO/HxxU8Pj7GBgGUmN0d8/8pELgOvuPguw6+Y6LrVIrAc/E9L7p23Wi/6xA4Bt9xyBqDb4gulPw3IY4xpByXlOPiudG8B8/x8DwX1/Xw4omHrufhealoEmLuuq4uuk7VkaqLJh96dXV4dfV4dXWk6utJ1dXj1tXhpFLRHxTxHyLGzf3REf9hktvO/TFTwaXAx6rswa+JfrM+BZwFbAb+DKyw1j4x2GPGI/j98f/dxZ/+/PuopspYXGtxjMXF4mBxTFS36BLtNxYcY6NaRmOjwM6EUW2hyVcrRgGgseSnzhNf5d4D8XauY1NoKJIJcwZmy8h9KMaZs9wHoTVEgUzBh2P81SLWgTD6sDRh4deM0YeXCV2c/NezXllrJC0hoZsldDLYgqwn8Veyxsvmv6p1vCyu6+N6PsaNAljrBuD6WDfAOhbfWELXEjoW3zGEjiXrQmBM/OFi8B2PwLr4eAR4BNaLZk6Two+vsyZFhjQZky7YTpE10XWfi+Phm9K/mnJsSH3g0xhk40u87fs0ZX0mZbPxpYdJmR5aenqYlOmmpaeHyd1dtPR0UZfJ9AaDtjeYs2F8bW3f22EQvZn67CsINgs//PO/LIYPrqqif9ancNtx+mR8cvsKMz8Dsjr57JCLdXrvDzHs+81vsDZg8vkXRF//ermvgaMxreuA62Adh8ymLey58UZa3vhGGk44HusYwvhr7Kg4IyrlsDbasjaIbwfxDHufbE8PT9zXRssUmHtYIzbIYIPoGwzrZwn9HsJMFzbbjZ/pIcz0kOnuxs9kCQnJWujBIWuji28dfBwC6xHiEOYCpDDOKoaj/w4jxBK4Nv4jL/pDL3QhcC1ZJyTrBljXIV3fBGkPm3Ih5WHTHngpHLcOx23AuHU4ph7j1uOaBoxpwAnTmMDF+G50HThY3yEMHKxvIHDAz2VH48xqGGdQwyiz6oYGJwQ3NHhh3zrx0bJEL11vDXv8EVqwnZ/FSbRt4nKdwmtDGH3jQ4iTSxXH3wCZKEXc59oYPz7exzE+xkTfDLmOjzFZHBPgOH70GCfofS7Xz3+zhBPE3yjF3zY58fO6PsbJfRsV4FmfxkyW+h6fhh6fhp6Q+oxPOhNQlw2oy4QY4n9X9BuI6JMb9lPPrrCFvaaZvbaJLieNnwI35eOlMzhegJvKkkpncFJZnFSA4wVYLyTEITBunBqJtgMMgXUJoumuWPp+A1TsW53cf4Qt/p9DNIMgum3yswx6f1/m99ne+3r/cIgu+W//rMHY3mOMLRwzerZoX67CKrrd+/bIPUfB28bGx4XRt3O5bazFhDb+VoHoZzdqORP/wW/j2yE2G0YdPYKQIBsluEI/nrAcJ7WC0EbdauL+5iEQ5LrYxB8RISZ6vxtDOIZWiib+A7nwj+W+29FtYy1uGMVDcRomPy/EjXurO8ahoWUSZ6366ajPZywq0e3hZOAZa+1z8QA3Am8BBg1+x8PWm37MgvahgoChg//CyTiD5wpK/wMi92M3tHhUG08c6n8mtnAf+Q+z/Gyg+L78F8K29wPPEs1Czz2f6TNO/Hy28EOy93b+OfvfHoH4M2AIvT88laqUql6hT2R/fBnRORTOIiv2QlgXhgnYq/3vLKa0czBxDXEQfa2OiRKR8Y+tNX1LG2xv9QIDVtVqnRJd//nu0k7wmAXw3KPRZQzadsFTzw1+f+4XhsHBsY3xL5feMubeX6LgWoOXv8/mfxH1P95Y8vsd23ucEz/GCcN8Vw4nmsaYD4DI1VAP2AbY2e8+ercH3K6M6JkdrElhjQeOl9/uvfagYNuaFNZJQeFt44JxscTXxsMat8j+gbdt/M1J/j7jAimsqQfj9D5XxV6FKrNxbX78/in8zWf6/M7p+zul+O8rCo4v3C5MFhV75XI/5f0fX+TYAY/vvT2Wxw/8NxQ83va7XWm5L/cGkftdWUyU2AyxcS1O1G0niG7boHc/0bc+Nr8dkpt2TJ8EQG5fVMuT695j87/Vo+Mg7Lff4oRRlrSWjCW+mAtsKri9GTil/0HGmMuBywHmz58/huFGx3G6cPxuBr6xC9/Mdsgflj6PsUX29T+uIAgt/EHqH1T2Bp82//dt73MNePp+s+kH3F2GPMlY2RKzNZX/4Bj/12J4Q51jSV/I2Gr9O3O/tEan8JHF/lnF3ve9f3gV7i98f/Uvv4ke4zd7ZCen4gf0jmxMvzHiu0w2pOGlHtyeMKr3jE+y78+fGbDPYPK10KHJ/SQbwtDkt/PZLIq9eibO4hZ7XUfy82H6XReX/8TpF4sMdnaDjTLgOfsEzTZ+qfoFSsVOMb/P9Hlhoyxrb74v1+fDhLn9WQyZgn9T39fK0v8D0xY9dtAt03/v4Pvz/0LjRMFwLgNpopAkNLlMZMH9Jv7GzzjxH3Vx+GJM3/v77MuFOCYaCwPGzZ9U4SvW++YueBUH7CMepyCrWmR/pDC06t1nTO9PoOn3HPn/TlMQ6A76Hiv2puh32xTZN+jtIimZ/B/Ixc5h8Ocb8KlU9a/2xzbegEfn30epsT71iFhrsWZf9QYs0ViC35I+ta211wHXQVT2MIbxRuXD/3lDtYcUERERkRo1lqZ/m4F5BbdbgcHXExURERERGWdjCX7/DBxhjDnEGJMG3g7cUp7TEhEREREpv1GXPVhrfWPMh4D/IyrJ/o619vGynZmIiIiISJmNaUK9tfZW4NYynYuIiIiISEWNfaFvEREREZEJQsGviIiIiBwwFPyKiIiIyAFDwa+IiIiIHDAU/IqIiIjIAUPBr4iIiIgcMBT8ioiIiMgBQ8GviIiIiBwwFPyKiIiIyAHDWGurN5gx24GNVRuw13RgxziMO1Hp9Ro5vWYjo9drZPR6jYxer5HR6zUyer1GZjxfrwXW2hn9d1Y1+B0vxpg11tpl430eE4Ver5HTazYyer1GRq/XyOj1Ghm9XiOj12tkavH1UtmDiIiIiBwwFPyKiIiIyAHjQAl+rxvvE5hg9HqNnF6zkdHrNTJ6vUZGr9fI6PUaGb1eI1Nzr9cBUfMrIiIiIgIHTuZXRERERCSZwa8x5l+NMU8aYx4xxtxkjJkyyHHnGGPWG2OeMcZ8ssqnWTOMMRcZYx43xoTGmEFnZBpjNhhjHjXGrDXGrKnmOdaSEbxeen/FjDHTjDG3G2Oejq+nDnLcAfseG+79YiJfj+9/xBhzwnicZ60o4fU6wxizN34vrTXG/PN4nGetMMZ8xxizzRjz2CD36/1VoITXS++vAsaYecaYu4wx6+Lfjx8uckzNvMcSGfwCtwNLrLUvA54CPtX/AGOMC3wDeD1wDLDCGHNMVc+ydjwGnA/8voRjX22tXVprbUuqbNjXS++vAT4J3GmtPQK4M749mAPuPVbi++X1wBHx5XLgm1U9yRoygp+vP8TvpaXW2s9V9SRrz/eAc4a4X++vvr7H0K8X6P1VyAc+Zq1dBJwKfLCWP8MSGfxaa39jrfXjm/cBrUUOOxl4xlr7nLU2A9wIvKVa51hLrLXrrLXrx/s8JooSXy+9v/p6C3BDvH0D8Ffjdyo1qZT3y1uA79vIfcAUY8zsap9ojdDP1whZa38P7BriEL2/CpTwekkBa22btfbBeHsfsA6Y2++wmnmPJTL47WclcFuR/XOBTQW3NzPwP0r6ssBvjDF/McZcPt4nU+P0/urrYGttG0QfksDMQY47UN9jpbxf9J7qVeprcZox5mFjzG3GmMXVObUJS++vkdP7qwhjzELgeOD+fnfVzHvMG49By8EYcwcwq8hdV1hrb46PuYIoFf+jYk9RZF9iW1+U8nqV4HRr7VZjzEzgdmPMk/Ffx4lThtfrgHp/wdCv2Qie5oB5j/VTyvvlgHtPDaGU1+JBoqVN9xtjzgV+QfR1qxSn99fI6P1VhDGmGfgZ8PfW2vb+dxd5yLi8xyZs8Gutfe1Q9xtjLgHeCJxpi/dz2wzMK7jdCmwt3xnWluFerxKfY2t8vc0YcxPRV4+JDEzK8HodUO8vGPo1M8a8ZIyZba1ti7/m2jbIcxww77F+Snm/HHDvqSEM+1oU/uK11t5qjPlPY8x0a+2OKp3jRKP31wjo/TWQMSZFFPj+yFr78yKH1Mx7LJFlD8aYc4BPAG+21nYOctifgSOMMYcYY9LA24FbqnWOE40xpskY05LbBs4mmvglxen91dctwCXx9iXAgOz5Af4eK+X9cgvw7njG9KnA3lwpyQFo2NfLGDPLGGPi7ZOJft/trPqZThx6f42A3l99xa/F9cA6a+2/DXJYzbzHJmzmdxjXAnVEX5sC3Getfb8xZg7w39bac621vjHmQ8D/AS7wHWvt4+N3yuPHGHMe8B/ADODXxpi11trXFb5ewMHATfHr6QH/Y61dPW4nPY5Keb30/hrgKuAnxpjLgBeAiwD0HosM9n4xxrw/vv+/gFuBc4FngE7gPeN1vuOtxNfrQuADxhgf6ALePsi3gAcEY8wq4AxgujFmM/AZIAV6fxVTwuul91dfpwPvAh41xqyN930amA+19x7TCm8iIiIicsBIZNmDiIiIiEgxCn5FRERE5ICh4FdEREREDhgKfkVERETkgKHgV0REREQOGAp+RUREROSAoeBXRERERA4YCn5FRERE5IDx/wGJH80aLZu6QQAAAABJRU5ErkJggg==
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">n_samples</span> <span class="o">=</span> <span class="mi">100_000</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span>
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<p><strong>CDF</strong></p>
<p>Below, I use my <a href="https://github.com/timvieira/arsenal/blob/5f70cf6b1465f3eea20acec58c58c3c055c611ab/arsenal/maths/rvs.py#L280">arsenal library</a> to fit the empirical distribution function and compare it to the analytical solution.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">Empirical</span>
<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">S</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span> <span class="mi">200</span><span class="p">)</span>
<span class="n">E</span> <span class="o">=</span> <span class="p">[</span><span class="n">Empirical</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">k</span><span class="p">])</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">m</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">])</span><span class="o">.</span><span class="n">T</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ncols</span> <span class="o">=</span> <span class="mi">4</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">),</span> <span class="n">nrows</span><span class="o">=</span><span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">N</span><span class="o">/</span><span class="n">ncols</span><span class="p">)),</span> <span class="n">ncols</span><span class="o">=</span><span class="n">ncols</span><span class="p">,</span>
<span class="n">sharex</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">ax</span><span class="p">)</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">ax_</span> <span class="o">=</span> <span class="n">ax</span><span class="p">[</span><span class="n">k</span><span class="o">//</span><span class="n">ncols</span><span class="p">,</span><span class="n">k</span><span class="o">%</span><span class="k">ncols</span>]
<span class="n">ax_</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">E</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">ts</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span>
<span class="n">label</span> <span class="o">=</span> <span class="sa">r</span><span class="s1">'$F_{(</span><span class="si">%s</span><span class="s1">)}(t)$'</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="n">ax_</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">F</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">'b'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">)</span>
<span class="n">ax_</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
</pre></div>
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<p>In the plots above, the red line is the empirical distribution funciton, and the blue dotted line is the analytical distribution function. In all cases, we see that the fit is good.</p>
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<p><strong>PDF</strong></p>
<p>Below, we check the PDF against histograms.</p>
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<div class="prompt input_prompt">In [10]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">),</span> <span class="n">nrows</span><span class="o">=</span><span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">N</span><span class="o">/</span><span class="n">ncols</span><span class="p">)),</span> <span class="n">ncols</span><span class="o">=</span><span class="n">ncols</span><span class="p">)</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">atleast_2d</span><span class="p">(</span><span class="n">ax</span><span class="p">)</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">ax_</span> <span class="o">=</span> <span class="n">ax</span><span class="p">[</span><span class="n">k</span><span class="o">//</span><span class="n">ncols</span><span class="p">,</span><span class="n">k</span><span class="o">%</span><span class="k">ncols</span>]
<span class="n">ax_</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">E</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'lightcoral'</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">E</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">E</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">ax_</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$f_{(</span><span class="si">%s</span><span class="s1">)}(t)$'</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="n">ax_</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span>
</pre></div>
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vTpPDxkCD2rjGGpKiQkhKfT0igqLeXJ5cvd8vqOtXMCsJBSSiUDlwHvGB2Lu0g+eU6wj5FCa71Ea52mtW6rtX7a9pjzgoeHtdaXaq27aa27aq1neDJobxoyZIhjBhdnaWlptG3b1oCIfN/HH39Ms5gYLmzdul77uahNG6xWK9999517AvMdHwCjz/H8GKC97esuwH19TXyA5NTZ5s2bR3x8PMPbtHHL/i5u147Nmzdz9OhRt+xP+C7Jp+qVl5fzpz/9iVatWvHIkCEu/U77xETuGTiQmVu28NNPP9U7hgDv2vcC8CegxrUW/GGCsaoknzzDflOhIADPSS4VUkK46tSpU3z55Zdc2bUrppD6HV79kpOJNpsDbpyU1nolFUsE1GQCMN025nAtEK+Uauad6IS3WSwWFi5cyIQJEwir5dpRNRnZrh0AX331lVv2J4S/mTdvHps2beKx3r2JqEUX80eGDiU2PNwxFro+nLv2BdLkL0qpccAxrfWmc23nDxOMCe9wtEgFYOusFFLCrRYuXIjFYuEq2yw49REWGsoFrVvz/fffuyEyv+LKkgOAf97xE5WtWLGC3NxcxrjxrnX3pk1p0qQJS5cudds+hfAX5eXlPPHEE3Ru3LjW3WXjIiK4vV8/5s2bx+7du+sVh72Q0kBxWb2W1vQ1FwLjlVL7gVnACKVUwPREEu5nL6QCLA8AKaSEm82bN4/WrVvT6zz90V01oGVLtm/fzqlTp9yyPz/hypIDFQ/KHT+/N3fuXGLDw7nIjd1GQkJCGDVqFMuWLcNqtbptv0L4g0WLFpGRkcEfhg6tU8+IewYOJNxk4j//+U+94rA43RwJpO59WuvHtNbJWusUKiYg+0ZrfaPBYQkfZs+F4rIyystr7A3ql3yukAqk5m9XBcp7zs3NZdmyZUyaNAmlqqsFam+gbarRH3/80S378xOuLDngskA5vlzlT++3rKyMRYsWMTotjXA3deuzG1ZQQE5ODhs3bnTrfoOdPx1f7uJv7/nZZ5+lVXw84+u4/EbjBg24vkcPZsyYwcmT5+qFfW72FikIzAkn3MHfji13CMb3XOp0/NsXMg4UPlVIRUREkJOTE1QHmdaanJwcIiIijA6l3hYtWkRpaSlXX3212/bZp0ULQkNCWOGG/up+ZBFws232voFAntb6SF12FGw55W/5tHr1anJycriso/uX3Rtmm7jim2++cfu+g1Ww5RP4X06tWbOGNWvWcN8FFxBazYxqrrqtXz9KSkocawXVRTAUUlrr77TW4+ryu5JPwcO5kAq0KdDdewu0npKTk8nOzibYxnpERESQnJxsdBj1Nm/ePJJjY+nw5ZfgphapqLAwejRrxo8HD7plf75AKfUJMBxIVEplA38HzFAxGyawBBgL7AEKqVjwuk6CMaf8KZ8WLFhAuMnECNvkEO6UGB1NlyZNWLFiBY899pjb9x+MgjGfwL9y6uWXXyY+Pp7JvXrVaz/dmjZl4MCBvPnmmzz00EN16mXh3LUvEAfZ15fkU/ColAsBNgW6TxVSZrOZ1NRUo8MQdZCfn89XX33F7b17u61bn93AVq14d8MGSkpKHIvh+TOt9fXneV4D97rjtSSnfJfWmoULFzKsTZt6L8Jbk6Gpqby/ejXFxcVBdwfUEySffNuxY8f4dM4cbu/Xj+iwsHrvb+rUqdx66618//33lRY7dpVzi1RBAI2RchfJp+ARyC1SPtW1T/ivJUuWYLFY6twn/VwGtGxJcVkZmzdvdvu+hTDK9u3b2bdvn0e69dkNTU2luLiYtWvXeuw1hPAVr91wA6Xl5dzWt69b9jfql1+IDQ+vc/e+YOjaJ4QrnAupQGuRkkJKuMX8+fNpHB1N/5Ytz79xLQ2w7VMuBkUgmf3ggyhgdFqax15jUOvWhISEyDgpEfDKy8v5YNMmhqSkkOamGUyjwsIY37kzn376aZ3uoleatU8KKRHELBaLY6FjaZESoori4mIWL17MZR071nsR3uo0iYkhOTaW1e+95/Z9C2GUpbt306dFC5rExHjsNeIiIujbty8rVqzw2GsI4Qu+/fZbDuTmckufPm7d7zXdu5Ofn8/nn39e69+t1CIlXftEECs5coRY2/WhtEgJUcWC226joKCAyz3Qrc+uT3Iymw4d8tj+hfCmo0ePsunQIUZ5sDXKbpDZzPr16wPuLqAQzj744ANiw8Pd3lV2cOvWJCcnM2NG7debla59QlQoLS8n1jZOVwopIar4PD2d2PBwBqekeOw1+rRowf5Tpzhx4oTHXkMIb1m8eDGAdwqp1q0pKyuTrrEiYJ0+fZpPP/2USd26EWE2u3XfISEhXNW6NUuXLKn1+Ue69glRodRqdUyqFGg39aSQEvVSVlbG0owMRqelEebmBUWd9W7RAoANGzZ47DWE8JbPP/+cFrGxdGva1OOvNaBVK0JCQli5cqXHX0sII8ydO5eioiJu6NnTI/u/oksXysrLWbhwYa1+T7r2CVGh1Gol1lZISYuUEE5++OEHThYVcZkHu/UB9GzWjBClWLdunUdfRwhPKy4u5uuvv2ZUWprblwqoTlxEBD179pRCSgSs6dOn06FDB/rYbri5W49mzUhJSOCTWi4Mby+kwkwmWUdKBDXnrn1B2SKllBqtlMpQSu1RSj1awzbDlVJblFI7lFLfuzdM4as+++wzIkJDudgDC4o6axAeTsekJNavX+/R1xHC077//nsKCgq80q3PbkBYGD/+8EOlrkZCBIKsrCxWrlzJjTfe6LEbE0opJnTuzPf79nHy5EmXf89isRAWFkaU2UyBFFIiiAV1i5RSygS8CowBOgPXK6U6V9kmHngNGK+17gJc7f5Qha/RWrNgwQIuatvWLYsfnk/vFi1Yv349FevVCuGfFi9eTGRoKEO9uBDloNatKS4rY+PGjV57TSG84ZNPPgFg3PHjHn2diZ0717p7n30R+eiwMOnaJ4JaqdUa1C1S/YE9WutMrbUFmAVMqLLNDcB8rfVBAK31MfeGKXzRpk2byMrKYpwHFxR11qt5c3Jycjh48KBXXk8Id9Na88UXXzA0NZVINw+KP5dBrVoBFa1hgUYp9Z5S6phSansNzyul1Eu2HhVblVK9vR2j8JyZM2fSLzmZ1IYNPfo6PZs3p1V8PPPmzXP5d+yFVJTZLF37RFCzWK1Ems2YlAq+FimgBZDl9HO27TFnaUCCUuo7pdQmpdTN7gpQ+K758+djUooxHTp45fV6NmsGVBRwQvijXbt2sW/fPq926wNoFB1Nh8REVq9e7dXX9ZIPgNHneH4M0N72dRfwuhdiEl6wY8cOtm7dytXdunn8tZRSjOvYkRUrVpCfn+/S79i79kVK1z4RxKxWKxowh4QQaTYHZSFVXafjqn2rQoE+wGXAKOBxpdRZVwpKqbuUUhuVUhuPe7gZXnjeZ599xuCUFBraVqv2tC5NmhAaGiqFlPBbnz74IACXermQAhjYqhWrV6+mvLzc66/tSVrrlcC5Bq5MAKbrCmuBeKVUM+9EJzxp1qxZhISEMLFLF6+83tiOHSkpKeGrr75yaXvnrn3SIiWCVant2A8zmYg0m4Oya1820NLp52TgcDXbLNVaF2itTwArgR5Vd6S1fktr3Vdr3TcpKamuMQsfkJ6ezq5duzy6CG9VEWYzXbp0kUJK+K2lu3fTpUkTkuPivP7aA1u1Ijc3lx07dnj9tQ3mSq8K4We01nz82msMTUmhcYMGXnnNgS1b0qhRIxYsWODS9s5d+2SMlAhW9kmOQk0mIkJDg7JFagPQXimVqpQKA64DFlXZZiEwRCkVqpSKAgYA6e4NVfiST+67D6i4Q+dNffr0YdOmTTLhhPA7ubm5rD14kFHt2xvy+hfYxkmtWrXKkNc3kCu9Kio2lF4TfuOnn34i8+RJrvRSaxRUXAhefvnlLF682HGX/Vycu/bJgrwiWNlzxRwSUnFTIdhapLTWZcB9wFdUFEdztNY7lFJ3K6Xutm2TDiwFtgLrgXe01tUO/BWBYVF6Ov2Sk2keG+vV1+3866+cOHGCrKys828shA/56quvsGrt9fFRdq0TEmjevHkwFlKu9KoApNeEP5k9ezahISFe7RUBMHHiRHJzc11al83RIhUWJoWUCFrOXfsignSMFFrrJVrrNK11W63107bH3tBav+G0zbNa685a665a6xc8FK/wAZmZmfx85AjjvXwCg4qZk0AmnBD+Z/HixTSMjKRvcrIhr6+UYkBCAisXLzbk9Q20CLjZNnvfQCBPa33E6KBE3WmtmT17Nhe1aUOCl8bo2vVfv56I0FA+//zz827rGCMlXftEEHPu2heULVJCVDV//nwALu/c+Txbul+XJk0wKSWFlPArVquVJUuWcHH79phCjPuzO7BVK7Lz8gJqCQGl1CfAj0AHpVS2Uup25x4TwBIgE9gDvA38zqBQhZts2LCBAwcOcEXXrl5/7aiwMIampvL555+ft4u5dO0TonLXvkAcIxVqdADC/3z66ad0b9qUlIQEr792pNlMx8aN2bx5s9dfW4i6Wr9+PTk5OYwaPtzQOAa0rOjhtnr1alrZxkz5O6319ed5XgP3eikc4QVz5szBHBLCWC8tvVHV6LQ0Hl68mPT0dDqf44ZiSUkJsbGxRJWVUVhaitYapaobsidE4Ko6a1+OtEiJYJadnc3atWsZb0BrlF2PZs1kwgnhVz7//HNMJhMj27UzNI6uTZrQICwsUNeTEkFAa82cOXMY0bYt8ZGRhsRgH+d4vu59zrP2lWtNSUmJN8ITwqc4d+0L1nWkhHCYceedAEwwuJA6duwYR47IMAfhHxa++y4XtGxp2IWfXajJRN/k5GCccEIEiHXr1pGVleW1taOq0yIuju5Nm563kLJ37YsymwECbmyIEK5w7toXHhoacDcUpJAStbJgxw66NmlC+8REw2Lo0axiLU3p3if8wYEDB9h57BijDZqtr6oBLVuybetW8vLyjA5FiFr76Pe/J8xkYoxB3frsRnfowI8//siJEydq3MZ51j6AgoICb4UnhM9w7toXbjJRXFxscETuJYWUcFlWVhbrsrK4wsA7gVDRPUkBa557ztA4hHCF/a61rxRSF7RqRbnWrF271uhQhKiV8vJyFu3caWi3PrvRaWmUl5ezdOnSGrdx7toHUkiJ4GTv2mc2mQgPDZVCSgSvefPmARjapQKgQXg47Ro14mfp2if8wOeff067Ro1oZ2ArrrM+ycmEKCXjpITfWbduHdmnTxt+DgLo2awZjaOj+WzatBq3KcnNhZ9/lkJKBDV7i1SodO0TwW727Nl0b9qUto0aGR0KPZo1Y6sUUsLHnT59mu+++85nWqMAYsLD6da0KT/88IPRoQhRK3PmzPGJbn0AISEhXJqWxoq9ex0XilVZrNaKRUhDKyZIDrQLSCFcUalrn62QCqTJwqSQEi7JzMxk3bp1XGnAuh3V6dGsGdmnT3P8+HGjQxGiRkuXLsVisTC2Y0ejQ6lkYKtWrFu3rsYLQCF8TXl5OfPmzWNku3bERUQYHQ4Al7ZvT15xMWvWrKn2+ZKyMsJDQwm3FVKB1qVJCFc4d+2z31SwBNAC1VJICZfMmjULwGcKqe62CSd++ukngyMRomYLFy6kUVSUY/0mX3FBq1YUFRVJ/gi/sXbtWrKzs5lo4IyxVV3Uti3mkBC++OKLs54rLy+ntLy80sWjtEiJYOTctS/MZAIC66aCFFLivLTWzJw5kwtataJVfLzR4QAyc5/wfaWlpSxZsoRRaWmYQnzrT+1A22K80r1P+IvZs2cTHh7uE9367GLCwxmcksLixYvPes5+8RhuMhEmLVIiiDl37YsIwFzwrbO78Enbtm1j586dTOrWzehQHOIjI0lJSGDTpk1GhyJEtVauXElubq5PXfjZNY2JoW3btrKelPALVquVuXPnMmbMGGJ9pFuf3ai0NNLT09ny4IOVHre3PoWFhgbkxaMQrnLu2hcWgK2zUkiJ83rvd7/DpJShi/BWp0ezZtIiJXzWggULiIiIYETbtkaHUq0BMTH8sGxZQA36FYFp9erVHDlyhMtt3YJ8ySjbRDJf7d5d6XFHIWUyObozBdLFoxCucu7aF4g3FaSQEudktVqZs3Url7RvT2J0tNHhVNKjWTMyMzM5deqU0aEIUUl5eTmfffYZo0aNItq2GKevGdiqFTmFheyucgEohK+ZPXs2kZGRjqLFl6Q2bEhaYuJZhZT9Lnx4gHZnEsJVVWftg8C6qSCFlDinb775hiP5+Vzfo4fRoZylp0w4IXzUhg0bOHToEFdddZXRodToAts4qZUrVxociRA1KysrY968eVx22WU0CA83OpxqjUpLY9X+/eTn5zsec+7aJ7P2iWBWaUHeYJ1sQik1WimVoZTao5R69Bzb9VNKWZVSk9wXojDS9OnTiYuI8Mk7gd1lwgnho+bPn09oSAhDd+wwOpQatWvUiMbR0VJICZ/2zTffcOzYMcb72IQtzkalpVFaXs6yZcscjzl37QvEu/BCuCrou/YppUzAq8AYoDNwvVLqrMEytu2eAb5yd5DCGPn5+cyfP58ru3QhwrYyuy9JjI4mOTZWJpwQPkVrzaeffsqw1FTiIyONDqdGSikGtW4thZTwabNmzSI2PJxL27c3OpQaDWzZkviICBYtWuR4zH4XPixA78IL4Srnrn3BOtlEf2CP1jpTa20BZgETqtnufuBT4Jgb4xMGmjVrFoWFhdzQs6fRodSoR/PmUkgJn7J161b27t3L5Z06GR3KeQ1q3ZqDBw+yf/9+o0MR4iwlJSXMnz+fyzp29MmbeXahJhOXpqWxePFirFYr8NuFYnhoKCEhIZhDQgLq4lEIV1W3IG8g3VRwpZBqAWQ5/Zxte8xBKdUCuAJ441w7UkrdpZTaqJTaePz48drGKrzsnXfeoVNSEn2Tk40OpUa9mjfnl19+ITc31+hQhAAqBsabTCa/KKQuTEkBZJyU8E1ffvkleXl5XOUjC8Gfy5i0NHJycvjxxx8Bp0LK1hoVERoaUBePSqmWSqlvlVLpSqkdSqkHz/9bIhiVlpaiAFNISEB2c3WlkFLVPFZ1vtwXgD9rra3n2pHW+i2tdV+tdd+kpCQXQxRG2LZtG+vXr+em3r1RqrpDwDf0bt4cwK9apc435lApNVwplaeU2mL7esKIOEXtaa2ZPXs2I0eOpJGPzXJZnU5JSSRERvL9998bHYoQZ5kxYwZJSUkMb9PG6FDOa0S7dphDQvj888+BynfhoaJlKpAKKaAMeERr3QkYCNxb3bAPIUpLSx1LAARiN1dXCqlsoKXTz8nA4Srb9AVmKaX2A5OA15RSE90RoDDGO++8Q1hYGNd27250KOfU01ZIrZo2zeBIXOPqmEPgB611T9vXP70apKizzZs3k5mZyeU+tmhoTUJCQrigVSsppITPyc3N5fPPP+fKNm0I9cH1o6qKi4jgwpQUFixYAFTu2mf/N5Duwmutj2itN9u+zwfSqdJbSQiouKngfEMBgq+Q2gC0V0qlKqXCgOuARc4baK1TtdYpWusUYB7wO631AncHK7yjoKCADz/8kCuvvNLn76o3jIoiJSGBzYcOGR2Kq1wdcyj80OzZswkNDWVcx45Gh+KyC1NS2Lt3L1lZWeffWAgvmTdvHhaLhet8/Gaes8s6dmT37t3s2rWr0qx9EJAtUg5KqRSgF7CuyuMynENQWlpKqG3WzaDs2qe1LgPuo2I2vnRgjtZ6h1LqbqXU3Z4OUHjfJ598Ql5eHlN8eMYxZ72aN+enw1UbSX3Weccc2lyglPpZKfWlUqpLTTuTE5XvKC8v55M332REaioJUVFGh+OyoampAHz77bcGRyLEbz766CM6dOjg6HXgD8Z26ADAggULKs3aBxVjpALp4tFOKdWAionGHtJan3Z+ToZzCKjStS9IW6TQWi/RWqdprdtqrZ+2PfaG1vqsySW01rdoree5O1DhHVprXn31VTo3bsxA24Kdvq538+Zk5eXhJ4WEK2MONwOttdY9gJeBBTXtTE5UvuOHH34g+/RprvGjO+gAXRo3plGjRlJICZ+xb98+Vq5cyaTmzX16jG5VLeLi6NevHwsWLDira1+YyRRQF48ASikzFUXUTK31fKPjEb7JuWtfRDC2SIngsnbtWrZs2cKd/fv7zQnMfsdyw4YNBkfikvOOOdRan9Zan7F9vwQwK6USvReiqIsZM2bQICzMcVfaX4SEhHDRRRfxzTffoHXVml4I7/vwww9RwHU9ehgdSq1NnDiRdevWsW/fPqByi1QgFVKq4gLhXSBda/0/o+MRvqu0tNRRSIUF6WQTIoj85447iA0P5+pu3YwOxWU9mzdHAevXrzc6FFecd8yhUqqp7SSFUqo/FXma4/VIhcuKi4uZO3cul3fqRFRYmNHh1NoFRUUcPHiQzMxMo0MRQe7U3//Oey+8wLA2bWgZH290OLV2sa2b+ccffwz8duEYFnhd+y4EbgJGOM0wO9booITvKS0txWwbI6WUIjw8PKAKqVCjAxC+Y//+/SxKT+f+QYNoEB5udDguiwkPp1Pjxqxdu9boUM5La12mlLKPOTQB79nHHNqef4OKmS/vUUqVAUXAdVqaCnza559/Tl5ens/PclkT+zipb775hrZt2xocjQhmqw4cICsvjydGjjQ6lDrpkJRUMW53926gooCCihapkwF08ai1XkX1XdWFqMS5ax9AeHh4QN1UkBYp4fDyyy8TohR39e9vdCi11i85mXXr1lFeXm50KOd1vjGHWutXtNZdtNY9tNYDtdZrjI1YnM+7775LcmwsQ2wFib9p16gRzWJi+Prrr40ORQS5GT/9RGx4OOP8YEHrmtzYq5fj+/AgmLVPiHNx7toHEBEREVC5IIWUAODUqVO8/fbbTOzcmRZxcUaHU2v9WrYkNzeX3ba7gEJ4y8GDB1m2bBk39OqFKcQ//6QqpbiobVuWL1+O1XrOddV9kixyHRhycnJYuGMH13bvTqTZbHQ4dXZV166OQfXOC5EG0l14IVzl3LUPKgqpQMoF/zzrC7d75ZVXyM/P58HBg40OpU76JScD8OOPPxociQg2H3zwAVprbujZ0+hQ6mVk27acOnWKjRs3Gh1Krcgi14Hjo48+osRq5eY+fYwOpV7iIyMZ36kTUWYzIU7r5wTSXXghXFVd175AygUppARnzpzhhRdeYFRaGt2aNjU6nDpp36gRcRERfjFOSgQOq9XKu88/z7DUVFISEowOp14uatsWpRRfffWV0aHUlixyHQC01rz11lv0bdHCb89DzqaNGcO8G290/BweeJNNCOES6donAt6bb77JyZMneWTIEKNDqbOQkBD6tmghhZTwqqVLl3IwN5db/PwOOkDDqCh6NWvGknfeMTqU2nLbIteywLVxVq5cSXp6OlMCIJegIp8GtW7t+FlapESwqtq1L9Amm5BZ+4LcmTNnmPbEEwxLTaV/y5bn/wUf1q9lS/6zciV5eXnE+eE4L+F/Xn31VZo2aODXA+OdjWjbludXrSI3N5d4/5l6ujaLXJ+xTdG8AGh/1i9p/RbwFkDfvn1lpkwvevnll0mIjOSqrl2NDsUjAm0dKRHYcnNzWb9+Pbt27eLXX3/FarUSHR1NmzZt6NmzJ126dHF5rVGLxUJoALdISSEV5F5++WVOFBbytxEjjA6l3i5o1Yry8nLWrFnDmDFjjA5HBLi9e/eydOlS/jR0aKVuC/7s4vbt+e8PP7Bs2TKuueYao8NxlUuLXDt9v0Qp9ZpSKlFrfcJLMYpzOHjwIAsWLOC+gQP9ch02V4SZTFgsFrTWfrPYvQguhYWFzJw5k48//piVK1c6ZkEODQ3FpDUlThMRNWnShKuvvprbbruNXk6zVFantLTUMekKVBRShYWFnnkTBpBCKojl5uby7LPPMiotjX5+3hoFFRNOmM1mvv/+eymkhMe99tprmEymgOjWZ9cvOZlGUVEsWrTInwopxyLXwCEqFrm+wXkDpVRT4KjWWssi177n9ddfR2vN7f36GR2Kx9hn8SspKSEiIsLgaIT4zZkzZ3j++ed5/vnnOXXqFB0SE/nLX/5C3wMH6Nq0KQ0jIwkJCaGkrIwDp06xITubr3/5hbfffptXXnmFSy+9lKeeeor+NSydU1paijn0t3IjPDyckydPeuvteZwUUkHs6aefJvfUKf7qPxdM5xQVFka/fv1YuXKl0aGIAJeXl8fbr7zCxI4daRYba3Q4bmMKCeHS9u1ZsmQJZWVlhIb6/ilCFrn2b2fOnOHNN99kwoQJtPKf7qS1Zl+Yt7i4WAop4RO01syaNYvf33UXR8+cYUyHDjwwcSIDW7WqaDWtsjh7eGgoaUlJpCUlMblXL3KLivhw0yZeWb2aAQMGcPPNN/Pcc8+RmJhY6fdKSkoIDQ93/BxoXftksokgtW/fPl566SWu79mT7s2aGR2O2wwdOpQNGzYEVLOx8D1vvvkm+RYL9w0aZHQobjemQwdOnTrF6tWrjQ7FZbLItf967733OHXqFPc0bGh0KB7l3CIlhNF+/fVXxo0bxw033EDz2FiW3X47n1x/PRe0bu1y19P4yEgeHDyYzQ88wO8HD+aTGTPo3LkzCxYscGyTkZHB/v37SXMqrgJtsgkppILUn//8Z0JDQwNibJSzPllZlJWVyXpSwmMsFgsv/OtfDE1NpWfz5kaH43Yj2rYlzGRi0aJFRociAlxpaSnPPfccF7Rq5feTHZ1PuFOLlBBG+vLLL+nWrRvfLFvGv0ePZvkdd9Qr/2LCw/n7xRfz7V130cxk4oorruD++++npKSEl19+mbCwMG7u3duxvbRICb+3fPly5s6dy4P9+9M8gLolAQxo2ZIQpfj++++NDkUEqOnTp3MkP58HArA1CqBBeDhDU1NZsGAB0vtNeNLs2bM5ePAgD1x4odGheFy4bbB9IN2JF/6lvLycJ554grFjx9LEZOL7qVO5Z+BATCHuKQW6Nm3K8jvu4J6BA3nllVcYPnw4H3zwAVd16kTjBg0c20VERARUHrj06SmlRiulMpRSe5RSj1bz/GSl1Fbb1xqlVA/3hyrcoaSkhHvvvZfUhAQeDMCTV2xEBD2bNWPFihVGhyICkMVi4emnn6ZPixaMbNfO6HA8ZkLnzmRmZrJp0yajQxEBymq18tRTT9G5cWNGtT9rJvqAIy1Swkj5+flcccUVPPXUU9xyyy0sv+MOOiQluf11wkJD+ffo0Xx49dVs27SJgoICpg4YUGmb8PDwgMqD8xZSSikT8CowBugMXK+U6lxls33AMK11d+ApbOtwCN/z91Gj2L17N8+OHUuE2Wx0OB4xol071q1bR25urtGhiAAzffp09u/fz6PDhwf0FMaXd+qE2Wxm9uzZRociAtSsWbPYvXs3fx4+nBA33RH3ZVJICaNkZWUxePBgFi9ezDNjxvB869ZEevj6b0KXLqy4807evvLKs7rAB2PXvv7AHq11ptbaAswCJjhvoLVeo7U+ZftxLRXreAgf8/PPP/PcDz9wTbduXBzAdwBHtm2L1WqVVinhVsXFxTz5hz/Qp0ULLg7g1iioGEQ8IiWFWW+/Ld37hNuVlZXxjwcfpEuTJlzesaPR4XiFTDYhjPDzzz8zcOBA9u3axdzrr2fqgAFeuwnYsXFjru7e/azHw8PDKSsrw+q0LpU/c6WQagFkOf2cbXusJrcDX1b3hFLqLqXURqXUxuPHj7sepag3i8XCLbfcQsPISKYF+BpLfZOTiY2N5auvvjI6FBFAXnrpJbLz8nhi5MiAbo2ym9ilC9l5eaxdu7ba5/OefJK8J5/0clQiELz33nvsycnhsSBpjQJpkRLet2jKFIb07w/5+Sy9/XZG+MgNQPv0/4FyU8GVv2DVXTFUe4tSKXURFYXUn6t7Xmv9lta6r9a6b5IH+maKmv3tb39jy5Yt/G/cOBpGRRkdjkeZTSaGNm/Ol3PmyN104RYnTpzg6aefZlT79gxr08bocLxibMeORISGMn369LOemzZtGtd/8okBUQl/d+bMGZ544gkGtmzJZUHSGgUy2YTwrjlz5jBpxgyS4+JYdscddGnSxOiQHIKxkMoGnOdFTAYOV91IKdUdeAeYoLWWFeN9yNdff82zzz7LrX36MK5TJ6PD8YqR7dqRnZfHrl27jA5FBIDHH3+cM2fO8I9LLjE6FK+Ji4hgfKdOfPzxx2ety7ZkyRK+2bOH8vJyg6IT/urZZ5/l6NGjPHXppUHRsmsnLVLCW1599VWuu/Za+iYn8+Vtt5EcF2d0SJWE2xbnDZRccKWQ2gC0V0qlKqXCgOuASguMKKVaAfOBm7TWu90fpqirgwcPcv3EiXRMSuLpUaOMDsdrLrWNAXNeGE6Iuli/fj1vvvkmU/v3p1PjxkaH41U39u7N6dOnmT9/fqXHd+3aRYnVytEzZwyKTPijvXv38szTT3Nlly70C/B1o6qSQkp4mtaaf/zjH9x3332M7tCB+TfdRHxkpNFhnSXoWqS01mXAfcBXQDowR2u9Qyl1t1LqbttmTwCNgNeUUluUUhs9FrFwWXFxMVdddRWWsjI+uvZaosLCjA7Ja1rExdEvOZl58+YZHYrwY2VlZdx99900a9aMx4YPNzocrxvcujUpCQm8++67jsdycnKwj3E9KDNjChdprXnggQcwm0z8K4hu6tmFy2QTwoOsViv33HMPTz75JJN79uSja67x+Mx8dRVoLVKhrmyktV4CLKny2BtO398B3OHe0ER9lJeXM2XKFDZu3MiMa6+lfWKi0SF53fhOnXj866/JzMykTZCMaxHuNW3aNH766Sc+vPpqYm130YJJSEgIN/XqxVPffEN6ejqdOnUiIyPD8fwBKaSEi+bNm8eSJUv416WXBtxC8K6IkBYp4SFFRUVc07s3X+zaxe8HD/b5CZHsLVKBkgvBMV1OEHrssceYM2cO/7zkkqAZF1XV5Z0rljur2i1JCFd8P3UqT/7970zq2pUJXboYHY5hpvTpQ0REBC+88AJApXGHB06dquG3hPjNsWPH+N3vfkfv5s25u8rinMEizDbZRKBcPArfcPLkSS699FIW79rFv0eP5u8XX+zTRRQEYdc+4X+mTZvGf/7zH+655x7uHzTI6HAMk5KQQI9mzZgzZ47RoQg/c/r0ae749FMSo6J4duxYo8MxVGJ0NNd16cL0997j+PHj7Nq1i/DwcBKjoqRrnzgvrTVTp04lPz+f1yZOJNRWUAQbWUdKuFtmZiaDBg1i/Y8/8u6kSdwzcKDRIbkk0Lr2SSEVYJ5//nkee+wxru7WjX8lJfn8nQlPu7pbNzZs2MD27duNDkX4Ca01d9xxB/tOnuSdSZNICPDlAlxxz8CBFJeV8dprr7Fr1y7S0tJIbdhQuvaJ83rllVdYsGAB//d//0fHIJusxZlZWqSEG61evZqBAwdy7NgxFtx8M1d27Wp0SC4rmzkTCJybClJIBQitNf/61794+OGHGd+pE69NnIgpSBY6PJfre/QgLCyMt99+2+hQhJ+YNm0ac+fO5YmRIxmckmJ0OD6hQ1ISYzt04H//+x+bNm2iY8eOtI6PlxYpcU7r1q3jkUceYXRaGredPm10OIZSShEREREwF4/COB988AEjRowg1mpl2eTJDGrd2uiQaiXQZrCUK+0AUFZWxr333svjjz/OjTfeyHuTJjnufgW7RtHRXJ6WxodvvUVRUZHR4QgfN2vWLP7yl79wdbduPHDhhUaH41P+NmIE+adPc/jwYVKOHqV1QgLZeXlYrVajQxM+6ODBg0ycOJFm0dG8PnFi0PeOAAgncC4ehfdZLBYeeOABbr31Vga2aMHyO+7wy4nEpJASPuXkyZOM6dSJ119/nQcGDeKlNm2Ctg96TW7u3Zu84mLmzp1rdCjCh33xxRfcfPPNDBkyhFcmTJALvyo6N2nC1d27A5CWmEir+HjKysvJzs42ODLha3Jzc7n88sspLCxk9g03SPdYm3CTKWAuHoV3ZWVlMWzYMF5++WXuGTiQ+Tfe6Ld5FWjjBaWQ8mMbNmygb9++fJ+ZyQvjxvHPSy8lRLrznWVISgqdkpKYNm0a5eXlRocjfNDixYu5auJEuiYlMePCCx13zERlj48Ywai0NIamptI6Ph6A/fv3GxqT8C35+fmMGTOG9O3b+XDChKBbxPpcwkNDA+biUXjP/Pnz6dGjB9u3b+f9SZP49+jRfn3DPNBmsJSrbj9UWlrK008/zaBBgyjNyWHJrbdyS9++Rofls0JCQvjjsGGkp6fLAr3iLO+//z4TJkygc+PGfHbTTcQF4XpRrmoZH8/sG26gSUwMrRMSANi3b5/BUQlfcerUKcaMGcOGDRt47+qruahtW6ND8ikRoaEBc/EoPO/UqVNMmTKFq666ipTISL6/9Vau8KNJJWoiLVLCUBs2bKB///787W9/Y3yHDqy65x76tWxpdFg+b0LnznRITOSpp56SMR0CqBhb+Mgjj3DbbbcxLCWFz2+5hfjISKPD8hstYmMxKcWePXuMDkX4gKysLIYOHcr6H3/kvSuv5PIgXb/wXMKkkBIu0Fozd+5cOrdqxcyPPuIPQ4fy1W230bZRI6NDcwt7j4/cAJmsSAopP5GVlcWtt95K//79OXr0KNOvuYb3rr5aLvxcZAoJ4dHhw9m+fTuvv/660eEIg2VmZjJ8+HD+97//cWf//sy6/npibGtbCNeEhYbSPjGRbdu2GR2KMNh3331Hnz59OLB7N3MnTw7qBazPJUK69onz2LFjB6NGjeKaa66hcYMGrLjzTv42YgRhAdTdPDosjF7Nm/O///2PAwcOGB1OvUkh5eMOHDjAXQMG0C41lY8/+oiHLryQ9bfcwvjOnY0Oze9M7NKFkW3b8tgjj5CVlWV0OMIAZWVlvPDCC/To0YNtGzfy1pVX8uzYsQF1kvKmrk2asHXrVqPDEAaxWCw89thjjBwxggStWXHnnQyX7nw1CpPJJkQN9u/fz2233Ub37t3ZsGEDz4wZwzd33knP5s2NDs3tlFK8e9VVlJWVMWnSpPOOs/3nP//Jtd27s+ePf/ROgLUkhZQP0lrzww8/cN1119GuXTve37iR63r0YNP99/OPSy6RO+d1pJTiuXHjKNeaW265hdLSUqNDEl6itWbhwoX07NmT3//+9wxs2pTV99zDNbZZ6ETddGnShP3795OXl2d0KMLLvv76a3r06MG0adO4sVcvVtx5J2lJSUaH5dOkRUpUtW3bNm655Rbat23Lx9OnM7V/fzbdeSdTBwzw6wklzqdNo0a8NnYs27dsoUOHDjzwwAPVzgCblZXFU089xZxt27jgtdfYuXOnAdGemxRSPmTXrl3885//pEOHDgwdOpSlCxcytV8/tjz4IC+NH09L2yxZou5SEhJ4duxYvvnmG+6//3601kaHJDyouLiYjz76iF69ejFx4kSKjxzho2uvZe7kyZJPbtClSRMA6d4XJLTWrF69mksuuYRLL72UkqNHmTt5Mi+NHy83+FwQHhrKr7/+ypkzZ4wORRioqKiIjz/+mBEjRtC9e3fmzJzJ7f36semBB/j36NE0io42OkSvuLxTJzbdfz/Xde3K66++StuUFG699VZ++uknxzYvvvgiWmvm3HADZywW3njjDQMjrp70ZzHQmTNnWLVqFV/8858s272bX3JyUMCFKSn8fuJEJnTuTHRYmNFhBpzJvXqxJyeH5998k4iICJ577jlMAXznJ9hYrVbWrFnD7Nmz+fjjjzl16hSdO3fmlQkTuK5794C+y+dt9kJq69atDB482OBohKfk5+czd+5c3nzzTdavX09iVBT/uvRS7ujXjwiz2ejw/MYVXbpwz8KF9OvXj7feeoshQ4YYHZLwkoKCApYtW8Znn33GggULyM/Pp3V8PP+4+GJu7t2bhn66JlR9tYiL46Xx43lkyBBeWbOGmTNn8sEHH9CnTx9Gjx7NW2+9xRWdOnFpWhoXtW3LwoULefHFF31qnUcppLykpKSEjIwMfvrpJzZu3Mi6devYvHkzVquVcJOJC1NSuLN/fy7v1IlmsbFGhxvwHh8xgqLSUl588UV2797Nm2++yZdffsncuXOZMWMGTWwXiML3lZeXs3v3blatWsU333zDsmXLyMnJISI0lLEdOnDzuHEMa9PGp/7wBormsbEkJCTIOKkAo7Vm//79fP3113zxxRd8/fXXFBcXk5aYyH/HjuX6nj3lJl8dXNujB01jYpg6fz5Dhw5l8ODB3HPPPVxxxRVE2iaO+vXXX3n88ce54YYbuOiiiwyOWNTV6dOn2bhxI6tWreLbb79lzQ8/YLFaiY+IYGKnTkzq1o0hKSmy9qdN64QEnr3sMv46YgSzt27lky1bmDZtGlpr7r/wQgDGdujAl4sWsWXLFnr16mVwxL9xqZBSSo0GXgRMwDta62lVnle258cChcAtWuvNbo7Vp1mtVk6cOMGhQ4fIzs7m4MGD7Nu3jz179pCRkcGePXsc025HR0fTMzGR3w8axKDWrRnYqhVRclLyqpCQEKaNGUO7Ro14/NtvSUlJcSzW+7e//Y23337bY68t+VQ3hYWFZGVlsW/fPn755Rd27drFtm3b+Pnnnzl9+jQAjaOjubhtW0ZddBGXtm9PA+lu5FFKKXr06GF4ISU5VXdFRUVkZmaSkZHBjh07+Omnn1i7di1HjhwBIDk2lik9ejCpa1f6JifLDYl6GtamDZsfeIAPN2/mrYwMJk+eTFRUFBdffDHNmjVj/owZHC8o4IsvvmDnzp0k2NZr80Xny7tAp7Xm+PHj7N+/n71795KRkcHOnTvZunUru3fvRmuNAro2bcrdAwZwcfv2XNCqFWbpFVGj+MhIpg4YwNQBAyguLSW3uJimMTEAjOnQgZCQEBYuXOhfhZRSygS8ClwCZAMblFKLtNbOI77GAO1tXwOA123/eoTW+qwv++Pl5eWUl5dX+t5qtTr+tX+VlZVRVlZGaWkppaWlWCwWLBYLJSUljq/CwkKKioooLCykoKCA/Px88vPzycvLIzc3l5MnT3Ly5ElOnDjBiRMnHBfidpGhoaQkJNC+USPGDRpEp8aN6da0Ke0aNcIkdyF8wh39+3NJ+/b85/vv6dOiBbtPnODNd97h3nvvpWfPnm5/PV/Mp/OpmmP23HL+3vnLOceq5ppznhUXFzvyq7CwkDNnzpyVXydOnODYsWMcOXLkrDUnYsPD6dS4MVenpdGrRQv6JyfTPjFRLvS8rENRETO2b2fhwoU0bNiQuLg4GjRoQEREBCEhIZhMJpRSxMfHE+qB2RH9MadqUlOuVf2qKbdKSkqwWCwUFRU5csueV7m5uZw6dYqcnBxHTmVnZ3Ps2DHH6yugbaNGDG7enP69ejE0NZU0ySm3iwoL456BA5navz8/7N/P5+npfLd6NT8WF9M6Pp5/jx7N1PnzeeCBB3juuedISkryuf8DF/POK6q7DqyaO87Xf865Y7FYKC0tdVz3FRUVUVxcTEFBgeO6Ly8vj7y8PMc56fjx4/z6668cPny40uQhioqWlS5NmnDlsGH0btGCfsnJskxNHUWYzTR16jqcGB3NgORkPv74Y4YOHUpqaipms9nxZTKZMJlMhIaGEhISglLK8a/9yxNcOav1B/ZorTMBlFKzgAmAc7JMAKbriiN5rVIqXinVTGt9pC5BPfzww44BZoDPTAhgUooG4eHEhocTGxFBw8hI2kZGMqBlSxI7dqRxgwY0i4mheWwsLWJjadyggc/98RNna52QwKsTJwKQW1TEnK1beeihh/j222898f/n9XzKyMigq9Nq6FXzyfln5xOSUSJCQ4mLiCA+IoKGUVG0i45mcPv2NI2JoUVcHCkJCaQmJEh++YgLWrXizXXrmGjLoZps27at0nHoRl7PqUsuuYTvv/++0mPnypuqOVb1e2/lXGRoKA2jokiMjqZxdDTdkpNp0aULqQkJtE9MpF2jRtKK60UhISEMa9OGYW3anPVc+rFjPDdjBjNmzHA85nxB+Lvf/Y6XXnrJm+FW5UreuWTOnDlMnjz5rMd9JW+gIncSIiOJj4wkKTqavg0a0LRPH5Lj4kiOiyM1IYHUhg2JlDGDHjV1wABunz+fkSNH1un37dcMVf8Nrcdi2a4UUi0A50V3sjn7Tl5127QAKp2klFJ3AXfZfjyjlMqoVbT1kwicqM8OrFqTV1xMXnExuG+633rH5QG+GBN4Ma7vv//+fH2XW9dx127LJ3BrTvnM/3lxWRnFZ85wtP4zW/nMe3Ijv31P3bp1q+5h+/upaz5B4Jyj6qJWx0NRWRmHTp/mkK0rrJ/z21yoK6cbXYkvv/zyiZdffvlcm9cnp1xx3rzzk3xy6TgqKiujKD+fw/n5XgjJwVePcV+NC84TW01FuNVqPd+N2RrzyZVCqro9V70N4Mo2aK3fAt5y4TXdTim1UWvd14jXPhdfjMsXYwLfjauW3JZP4L6cCpDPthJ5T77PTe8nIM5RdRFox0NtyHs3/L2fN6f8IZ985LOslq/G5qtxgTGxuTJQJxto6fRzMnC4DtsIISSfhHA3ySkhvE9ySghcK6Q2AO2VUqlKqTDgOmBRlW0WATerCgOBvLr2PRciwEk+CeFeklNCeJ8reSdEwDtv1z6tdZlS6j7gKyqmuHxPa71DKXW37fk3gCVUTCu7h4qpZW/1XMh15qvNy74Yly/GBL4bl8t8OJ/8/rOthrwn31fv9+PDOeUNgXY81Ia8dwPVlHcGh1UXhn+W5+CrsflqXGBAbMroGbqEEEIIIYQQwt/IYkZCCCGEEEIIUUtSSAkhhBBCCCFELQVlIaWU+oNSSiulEn0glqeUUluVUluUUsuUUs2NjglAKfWsUmqXLbbPlFLxRscEoJS6Wim1QylVrpTyyek3/ZWvHov14avHcV0F0vGvlBqtlMpQSu1RSj1qdDz+KNCO79oKpHxwleRN/bh6zCil9iulttnOhxt9LDavHgNKqYZKqa+VUr/Y/k2oYTuvfWbn+wxsEwu9ZHt+q1Kqt6diCbpCSinVErgEOGh0LDbPaq27a617Al8ATxgcj93XQFetdXdgN/CYwfHYbQeuBFYaHUgA8tVjsT589Tiuq4A4/pVSJuBVYAzQGbheKdXZ2Kj8UqAd37UVEPngKskbt6jNMXOR1rqnF9clOm9sBh0DjwIrtNbtgRW2n2vi8c/Mxc9gDNDe9nUX8Lqn4gm6Qgp4HvgTNSxw6m1aa+cl5qPxnbiWaa3LbD+upWKNCMNprdO11r64Orrf89VjsT589TiuqwA6/vsDe7TWmVprCzALmGBwTH4n0I7v2gqgfHCV5E09+fIx42JsRhwDE4APbd9/CEz08OudjyufwQRguq6wFohXSjXzRDBBVUgppcYDh7TWPxsdizOl1NNKqSxgMr7ZCnAb8KXRQQjP84NjsT7kOPYdLYAsp5+zbY+JupPjO/BJ3niPBpYppTYppe4yOhgnRhwDTezr7tn+bVzDdt76zFz5DLz2OZ13HSl/o5RaDjSt5qm/An8BLvVuROeOSWu9UGv9V+CvSqnHgPuAv/tCXLZt/gqUATO9EZOrcYm68dVjsT589TiuqyA5/lU1j/l9C6gnBNrxXVtBkg+ukrxxgZuOmQu11oeVUo2Br5VSu7TW9e5C6obYPHIMnOfa2VUe+cyq4cpn4LVcCbhCSmt9cXWPK6W6AanAz0opqOgCsVkp1V9r/asRMVXjY2AxXrp4PV9cSqkpwDhgpPbigmO1+LxELfnqsVgfvnoc11WQHP/ZQEunn5OBwwbF4tMC7fiurSDJB1dJ3rjAHceM1vqw7d9jSqnPqOhOVu+iwA2xeeQYOFdcSqmjSqlmWusjtu5xx2rYh0c+s2q48hl4LVeCpmuf1nqb1rqx1jpFa51CxYfc29NF1Pkopdo7/Tge2GVULM6UUqOBPwPjtdaFRscjPM9Xj8X6kOPYZ20A2iulUpVSYcB1wCKDY/I7cnwHHckbL1BKRSulYuzfU9GTabuxUTkYcQwsAqbYvp8CnNVy5uXPzJXPYBFws232voFAnr17orsFTSHlw6YppbYrpbZSceA9aHRANq8AMVQ0z25RSr1hdEAASqkrlFLZwAXAYqXUV0bHFEB89VisD588jusqUI5/2wQJ9wFfAenAHK31DmOj8ksBdXzXVqDkg6skb+qvpmNGKdVcKbXEtlkTYJVS6mdgPbBYa73UF2Iz6BiYBlyilPqFilmvp1WNCy9+ZjV9Bkqpu5VSd9s2WwJkAnuAt4HfeSIWABWAPQGEEEIIIYQQwqOkRUoIIYQQQgghakkKKSGEEEIIIYSoJSmkhBBCCCGEEKKWpJASQgghhBBCiFqSQkoIIYQQQgghakkKKSGEEEIIIYSoJSmkhBBCCCGEEKKWpJASQgghhBBCiFqSQkoIIYQQQgghakkKKSGEEEIIIYSoJSmkhBBCCCGEEKKWpJASQgghhBBCiFoKNeqFExMTdUpKilEvL0S9bNq06YTWOsnoOJxJTgl/JfkkhHv5Wk5JPgl/dq58MqyQSklJYePGjUa9vBD1opQ6YHQMVUlOCX8l+SSEe/laTkk+CX92rnySrn1CCCGEEEIIUUtSSAkhhBBCCCFELUkhJdzm0KFD/Prrr0aHIYRhDh8+zOHDh40OQ4iAVVBQQHp6utFhCOHzsrKyOHLkiNFhBDzDxkhVp7S0lOzsbIqLi40OxasiIiJITk7GbDYbHUq93HzzzSQkJDBv3jyjQxHCEHfddRdaaxYvXnzebYPx712g/K0Txnnttdd48sknOX36NCEhv90LDsZ8AskpUbMJEyaQkpLC/Pnz6/T7wZhTdcknnyqksrOziYmJISUlBaWU0eF4hdaanJwcsrOzSU1NNTqcejlx4oTRIQhhqJMnT6K1dmnbYPt7F0h/64Rxjh07RkFBAaWlpYSHhzseD7Z8AskpUbPjx4/z008/ERcXV+d9BFtO1TWffKprX3FxMY0aNQqK/zA7pRSNGjUKiIq/uLiY0tJSo8MQwjAWiwWLxeLStsH29y6Q/tYJ4xQUFACclWfBlk8gOSVq9v333wPU65os2HKqrvnkUy1SQND8hzkLlPcshZQIdhaLxeUWKQic3HdVsL1f4X6FhYXA2YUUBOfxFYzvWZzfd999B0DRvn312k+wHV91eb8+V0gJ/1VUVCSFlAhqtS2khBC1U1OLlBDiN99++y0AFqvV4EgCn0917RP+TVqkfqOUek8pdUwptd3psYZKqa+VUr/Y/k2o4XdHK6UylFJ7lFKPei9qUV8Wi4WSkhKjwxAiYNlbpORcI0T1jh07xs6dOwEoKy83OJrAJ4VUDR555BE6d+7M/fff79L2RUVFDBs2DKut+s/Ozmb27NlAxcXV0KFDKSsr81i8vqDYNgBYAPABMLrKY48CK7TW7YEVtp8rUUqZgFeBMUBn4HqlVGfPhircpaSkxC/vlMvfO+Ev8nbsAHy7RSpY8kkp9Xul1A6l1Hal1CdKqQijYxKwcuVKAFISEgKmRcqXc0oKqWpkZmayevVqdu7cycsvv+zS77z33ntceeWVmEwmAFasWMHmzZsBCAsLY+TIkY7/xEBktVopLS+XQspGa70SOFnl4QnAh7bvPwQmVvOr/YE9WutMrbUFmGX7PeEHajPZhK+Qv3fCnxTZzjG+mmfBkk9KqRbAA0BfrXVXwARcZ2xUAipapADaNmxIaQAUUr6eU1JIVZGRkcGwYcM4cOAAvXr1cvTHPp+ZM2cyYULF9e6qVat4+OGHmTdvHj179mTfvn1MnDiRmTNnejJ0Q9m7M0khdU5NtNZHAGz/Nq5mmxZAltPP2bbHqqWUuksptVEptfH48eNuDVbUnr8VUsH0907ungeGQh8upIIpn2xCgUilVCgQBchq5D7Afh0WFRbm94WUP+SUTDZRRYcOHZgyZQopKSnccccdLv2OxWIhMzOTlJQUAAYPHky/fv3473//S9euXYGKFpsNGzZ4KmzD2aeL9MXuB36muiljapy9QGv9FvAWQN++fWWWA4NZCgv9arKJYPl753T3vLPWukgpNYeKu+cfGBqYqLUCWwHli4VUsOQTgNb6kFLqv8BBoAhYprVe5ryNUuou4C6AVq1aeT/IIGXPjeiwML/v2ucPOeW2Qkop9XvgDiou+rYBt2qt67y4wUMPPcSWLVvcFF2Fnj178sILL5x3u23btjkq2QcffJB///vfrF+/nscff5wuXbpw3XXXERMTw8aNG5k6dSonTpwgPj6+0j4yMjLo0KGD42eTyURYWBj5+fnExMS48235BHshJS1S53RUKdVMa31EKdUMOFbNNtlAS6efk5G7fH7DYrWitUZrXatpVH3t711ERASPP/44p0+fpm/fvnTt2jUQ/t7Z756XInfP/ZYrXft8LZ82bdrEzJkzKSsrc3RP8vd8sk2WNAFIBXKBuUqpG7XWM+zbyI0+Y9ivw6LNZrdNNuFrOXXixAnuu+8+EhMTSUtL45JLLjEsp9zStS/Q+sru2LGDLl26cPLkSZRSREVFoZSiQYMGFBcXk5ycTJ8+ffjhhx8AiIyMrLSAV05ODnFxcZjN5kr7LSkpISIiMHuTSCHlkkXAFNv3U4CF1WyzAWivlEpVSoVRkUeLvBSfqAer1Uq51mjb9/6iur93Cxcu5NChQ5jN5oD4e6e1PgTY754fAfKq3j0H6SrrD3y5ax9Un09DhgzhjTfeYNy4cUyZMsXv88nmYmCf1vq41roUmA8MMjgmwW+5ERUALVJQfU7t3r2byy67jPfee4+dO3camlPu7Nrn1rt9rlSpnpCfn4/ZbCYqKoq1a9fSuXPFhGlDhgxh2LBhHD16lIcffpiZM2cSERHB0aNHadKkCVarleLiYiIiIti3bx/NmzevtN+cnBySkpLO+o8MFFJIVaaU+gQYDiQqpbKBvwPTgDlKqdupuKC72rZtc+AdrfVYrXWZUuo+4Csqbki8p7XeYcR7ELXjfGFnsVgIDXX9z6uv/b3LyMjgggsuYOrUqUyaNImRI0f69d87V+6eg9xB93Vaa0chda5zja/lk93HH3/MO++8A+DX+WRzEBiolIqiomvfSGCjsSEJsJ1/QkIIN5ncNkbK13KqV69ePP3008yePZubbroJMC6n3NIiFUh3+7Zv3+7oQ3ny5ElH82BISMVHlZCQ4JhYISEhgdOnTwNw6aWXsmrVKgA6duzIiRMn6Nq1K2vWrAEqFkcbO3asN9+KV0khVZnW+nqtdTOttVlrnay1fldrnaO1Hqm1bm/796Rt28Na67FOv7tEa52mtW6rtX7auHchaqNqIeUPavp7l5ycTEJCxTJn9lmP/Pzvndw9DwBFRUWO730xx2rKJ4CDBw8SFxdHbGws4Pf5hNZ6HTAP2EzFcI4QbDchhLFKS0sxh4QQajJh1dqvekhUVVNOvf/++zz55JN88803LF68GDAup9zVtc/5bl9zIFopdWPV7bTWb2mt+2qt+yYlJbnjpd3uggsuYO7cuQCkpaWxf/9+AObPn8/UqVO56aabuO+++wA4dOiQYwDlfffdx4cfVsxs3aBBA9avX8/27dsZNKjiXP3xxx9z1113efndeI8UUiLY+WMhVdPfuyuvvJKvvvqK+++/n6FDhwJ+//fOcfdcVQxeGwmkGxyTqCX7YrzgmzlWUz4BvPvuu9x6662On/08nwDQWv9da91Ra91Va32T1lpWI/cBFosFs8lEmO0mmD9fl9WUU6NHj+all17i7rvvdkwqYVROuatrn+NuH4BSyn63b8Y5f8vHdevWjddeew2ouLC48sorHc+dOXOG2NhYwsPDgYpmxosuugir1eq4g2tnsViYOHFipYFugcZeSJWXl1NeXu5owRMiWPhjIeXM+e9dVFQU7777ruM5f/97p7Vep5Sy3z0vA35C7p77Heepj309x5zzCeDJJ590fO/v+SR8m8ViIcxkwmy7DistLfW18XV14pxTXbt2Zd68eY7njMwpd13tBuTdPqUUkydPrnQXzO7w4cP88Y9/rPTYbbfddtZ/GFQs/nXzzTd7LE5f4Dywz5/vfghRV84Xdvbuv/4k0P/eyd1z/+frLVLOAj2fhO8qLS3FbDJhth1Pvp4rrvLVnHJLi1Qg3+0bMmRItY+npaV5ORLfVrWQst8VECJY+HuLFMjfO+Hb/KlFCiSfhDECqWtfVb6YU26btU9r/XcqZiYTQci5kJJFeUUwcm6F8oeLPCH8jT+1SAlhFOfJJkByxdNkIItwC+naJ4JdILRICeEpxcXFPPnkk/Xq9upvLVJCGME+RipMCimvkEJKuIXztLRSSIlgJIWUEDX74Ycf+Mc//sGPP/5Y5304t0jJeUaI6tm79jlPNiE8x+cKKa2Dbw3EQHjP0iIlgl1dCqlAyP3aCLb3K35z5swZoPK5orbO1yIVjMdXML5ncW7unGwi2I6vurxfnyqkIiIiyMnJCar/OK01OTk5fj81pRRSItjVtpAKtr93gfK3TtSNvQiqT9e+c42RCrZ8AskpUb2qXfvqek0WbDlV13xy22QT7pCcnEx2djbHjx83OhSvioiIIDk52egw6kUKKRHsaltIBePfu0D4Wyfqxh2F1LlapIIxn0BySpzN0bWvni1SwZhTdcknnyqkzGYzqampRoch6kAKKRHsaltIyd87EUzc2SIVERp6Vo5JPglRobS0lIiQEEchVddrMskp1/hU1z7hv6SQEsFOJpsQombuapEKM5mINJslx4SoQdXJJiRXPEsKKeEWUkiJYCeFlBA1c1eLVJTZTJjJJDkmRA1KS0tl+nMvkkJKuIUsyCuCnfPJqj4Xi0IEIne1SEWZzZhNJrlhJ0QNqo6RklzxLCmkhFtIi5QIdtIiJUTN3NYiFRaGOSREckyIGrhrsgnhGimkhFtIISWCnRRSQtTMXS1SkdK1T4hzKi0txRwSUu/pz4VrpJASblFcXEyorKJ9XkqpDkqpLU5fp5VSD1XZZrhSKs9pmycMClfUgvMFolzkCVGZO8dImaWQEqJG9nWkQmWyCa/wqenPhf8qLi6mQVgYucXFUkidg9Y6A+gJoJQyAYeAz6rZ9Aet9TgvhibqSVqkhKiZO8dIlVqtkmM+QCkVD7wDdAU0cJvW+kdDgxKOrn0y2YR3SIuUcIvi4mIahIcD0iJVCyOBvVrrA0YHIurPYrEQohQmpeTE5YOUUvFKqXlKqV1KqXSl1AVGxxRM7IWUczfw2iosLJSufb7lRWCp1roj0ANINzgega1rn1MhJddkniWFlHCL4uJiYsLCAEnaWrgO+KSG5y5QSv2slPpSKdWlph0ope5SSm1USm0MptXHfZG9O4Vc5PksuegzkLtapKLDwqRrnw9QSsUCQ4F3AbTWFq11rqFBCcCpa5+0SHmFWwspueMXvKRFqnaUUmHAeGBuNU9vBlprrXsALwMLatqP1votrXVfrXXfpKQkj8QqXGM/eclFnu+Riz7juWuMlL1FSs4zhmsDHAfeV0r9pJR6RykV7byB3OjzvvLycqxWq7RIeZG7W6Tkjl+Qso+RAklaF40BNmutj1Z9Qmt9Wmt9xvb9EsCslEr0doCidpz7pUsh5XPOe9EHcuHnSe5eR0pyzHChQG/gda11L6AAeNR5A7nR53326y9zSAhmmWzCK9xWSMkdv+Dm3CIlC/K65Hpq6NanlGqqlFK27/tTkac5XoxN1IF07fNp573oA7nw8ySZtS/gZAPZWut1tp/nUZFjwkD2vAgzmTCFhKCQQsrT3NkiJc28QayoqIhoaZFyiVIqCrgEmO/02N1KqbttP04CtiulfgZeAq7TWmvvRypqw3kRRDlx+Ry56DNQeXk5RUVFQN0LKYvFQllZGVFhYXKzwgdorX8FspRSHWwPjQR2GhiS4Lfrr1CTCaUUZukG63HuLKSkmTeIyWQTrtNaF2qtG2mt85wee0Nr/Ybt+1e01l201j201gO11muMi1a4yrlFqj533YX7yUWfsQoLCx3f1zU37PuIlBYpX3I/MFMptZWKZT3+z9hwhHOLlP1fyRXPcuc6UtXd8Tur64QIPGVlZVitVplsQgQ1eyFl/174HPtFXxiQCdxqcDxBw96tD+peSNn3ESXTn/sMrfUWoK/RcYjf2PPCbDsXSYuU57mtkNJa/6qUylJKdbAtOip3/IKEfV0QmWxCBDN71z7pk+6b5KLPOO4opOxdA2UdKSFq5jzZhP1fyRXPcmeLFMgdv6BkL6RkjJQIZhaLhXBpkRLiLPZCKkSp+hdSoaHStU+IGkjXPu9zayEld/yCk72QijSbCQ0NlUJKBCVHi5RScuISwom9kIqPiKh3IRUh60gJUaOqXftCJVc8zt0tUiII2QupiNBQzGazJK0ISiUlJUTYCqlCKaSEcLAXUgmRkfUvpKRFSogaObr2SYuU17h7QV4RhKSQEuK3ySbkIk+IytxZSEWazZhDQigvL8dqtbotRiECQdWufTLZhOdJISXqzV5IhUshJYKYvWuf3AEUojJ3FlLhoaEyO6YQNajaIiWTTXieFFKi3qq2SJWVlRkckRDe57yOlJy4hPiNW1ukbF37QAopIapyjJGyzdon5yPPk0JK1Ju0SAnxW4uUdO0TojLHZBO2QkprXet9VJ1sAqSQEqKqql37ZLIJz5NCStSbc4uUzNongpW0SAlRPecWKahbAeSYHVa69glRI/v1V6hMNuE1UkiJepMWKSEqF1J17b4kRCAqKCjAZDLRIDwcqNuivM4tUvaufXKuEaKy6taRkjzxLJn+XNSbzNonhKwjJURNCgoKiI6OdixYXZ9CSlqkhKjZWV37ZLIJj5NCStSb/QQXJoWUCGIWi4Xw0FDH90KICo5CypYfdS2kQkNCCLWNQwTJMyGqcnTtk8kmvEYKKVFvhYWFAESbzVJIiaBlb5Gyf6+1RillcFRCGM9eSIXVs0Uq0laISSElRPWqXUdKupp7lIyREvVmH0gcFRYmhZQISlarFavV6hgjpbWWxUKFsHFXi5T996VrnxDVs19/ORdSkieeJYWUqDf7QOIwk0kKKRGUnE9e4XKRJ0Ql7mqRijCbASmkhKiJPSdCZbIJr5FCStSb/SSplJIFeV2glNqvlNqmlNqilNpYzfNKKfWSUmqPUmqrUqq3EXEK19kvDM0yfkOIs5zOyCD811+JqGeLlHTt8y1KKZNS6iel1BdGxyIqnNW1Tyab8DgZIyXqraCggKioKABpkXLdRVrrEzU8NwZob/saALxu+1f4KOeTl6rymPAdSikTsBE4pLUeZ3Q8waLAYqFxgwaE1aOQKi4uPqtFSs41hnsQSAdijQ5EVJCufd7n1hYpuTsRnOwtUiCFlJtMAKbrCmuBeKVUM6ODEjVzLqSk25FPs1/4CS8qLC0l2myu9/Tn0iLlO5RSycBlwDtGxyJ+Y7FYUEphss3aZ5aufR7n7q59cpIKQlJI1ZoGlimlNiml7qrm+RZAltPP2bbHzqKUuksptVEptfH48eMeCFW4wn5BJ137fJdc+BmnqLSUqLCwek82IWOkfMoLwJ+A8po2kPOT91ksFsLCwhw/S4uU57mtkJKTVPByLqRCQ0OlkDq/C7XWvanownevUmpoleermzNbV7cjrfVbWuu+Wuu+SUlJ7o5TuEhapPzCC8iFnyGKSkuJNJvdN/257W675JgxlFLjgGNa603n2k7OT95XWlqK2XbDASpyRWaR9Sx3tki9gJykgpK0SNWO1vqw7d9jwGdA/yqbZAMtnX5OBg57JzpRF9W1SNXlYlF4hlz4GUdrTYHFQpTZ7LYWKWn1NdyFwHil1H5gFjBCKTXD2JAEnN0iJTf2PM8thZScpIKbFFKuU0pFK6Vi7N8DlwLbq2y2CLjZNnvfQCBPa33Ey6GKWpAWKZ8nF34GKS0txaq1W1qkZB0p36C1fkxrnay1TgGuA77RWt9ocFiC6rv22R8XnuGuFik5SQWxM9nZhGVmAlJIuaAJsEop9TOwHlistV6qlLpbKXW3bZslQCawB3gb+J0xoQpX2U9S4VJI+SS58DNOYWEhAJFOLVLFxcW13o9z1z7JMSGqd1bXPpnh0uPcMv251vox4DEApdRw4A9ykgoehaWlRNm7XEghdU5a60ygRzWPv+H0vQbu9WZcon6cu/ZVfaw2XnnlFZYuXcoXX8jEpyIw2AupKDe0SEnXPt+jtf4O+M7gMISNdO3zPlmQV9RbocVClC1xZUFeEYzc1bVvzZo1LF68mF9//dWt8YnfaK2/kzWkvKe6Fqm6riNVtUVKbtoJUVlpaWmlQirUNjGL5IrnuL2QkpNUcNFaU2BbIwSkRUoEJ+dCyn6xWJdC6vTp0wD88MMP7gtOCANVKqTq2CJltVqxWCzSIiXEeVgslkpd+6RFyvOkRUrUS0lJCeVaV2qRkkJKBBt3rSOVl5cHwMqVK90XnBAGcu7aFxISQmhoaK0LKfuYKnuLlCkkBJNScnEoRBXStc/7pJAS9VJQUABQaYxUWVkZFcN8hAgO9gvD+nbts7dIfTt3rvuCE8JAzoUUQHh4eK0LqaKiIgBHixTIQqMiMK1cubJewyPO6ton3WA9TgopUS+OQsqpRQqQcVIiqNjvmIeFhjq69tkvIGvD3iK18+hRTp486b4AhTCIWwup0N/mxwqTQkoEmL179zJs2DDmz59f531I1z7vk0JK1EvVFqmy774D5O6HCC72AiguPJy4iAgAcnNza72f06dP061bNzSwevVqN0YohDHs54hI2zkiIiLCLYWUtEiJQHP06FEAsrOz67yPs9aRkskmPE4KKVEvVQspmSFGBCN7IRUTHk5MWBghSnHq1Kla7UNrzencXIY2aIACNm7c6IFIhfAu58kmwH1d+8JMpjrN/ieEr7KfM44fP17nfdS0jpTcdPAct6wjJYKXvZCKtnftk/64Igjl5eURbTY7+qPHRUTUukWqsLAQq9Y0jo4mJjy8Ti1aQvgad3TtqzrZBFScc86cOeOmKIUwnjsKqZomm5BrMs+RFilRL2dNNiEtUiII5eXlObr0QUUhVdsWKXurVmxEBLHh4Y6JJ4TwZ55qkYqvw80KIXyZ/Xg+vGpVnfdRdbIJaZHyPCmkRL1UnWxCZogRwSg3N5dYp0KqLhd59sIpJjycGCmkRIBwZyHl3CJVl5sVQvgy+/GcY7uuqouqk01IIeV5UkiJerGfJKOrtEjJrH0imOTl5REbHu74uV4tUlJIiQBSWFhIuMmEyXZucFuLVGSktEiJgOLo2lfPQsq5RcreW0i6wXqOFFKiXs6a/lxapEQQysvLq9wiFRlZ60JKWqREICosLHScH0BapISoif3GQE4dls6wq9q1r3GDBgD8+uuv9YpN1EwKKVEvMkZKiLNbpOrStc95jFSD8HDy8/PdGaIQhigsLHScH6B+hVS4UyFlzzFZ/F0ECvuNgdMlJXWekbJq176Y8HAahIVx+PBht8QoziaFlKiXs6Y/lxYpEYSqjpGqy91yaZESgaiwsNAxPgogJDOTwoMHa7UPR4uU037iIiIoKytznIOE8HfON99OnDhRp31U7doH0KRBA44cOVKf0MQ5SCEl6qWgoIDI0FBCbC1R0iIlglHVWfviIyMpKSlxTNvs6j5AxkiJwFK1kIoMDaWwlueH6hbkjY+MBOq28LUQvujUqVOEKAXUfQr00uJiytetq/RY05gYaZHyICmkRL0UFBRU6v8uY6TOTSnVUin1rVIqXSm1Qyn1YDXbDFdK5Smltti+njAiVuGaEls3jKpd+4BatUpVbZHKz8+nvLzcvcEK4WUFBQWVuvY1jYnh11oe2451pKq0SEHtckwIX3bq1ClaxccDdSuktNZYrFbHdZhd05gYaZHyICmkRL1UPUmapEXqfMqAR7TWnYCBwL1Kqc7VbPeD1rqn7euf3g1R1Ibz2Ca7ulzk5eXl0SAsDFNICDG2okxmWnIPV25gCM+o2iLVKj6eEqu1VoPfi4qKMJvNjvML/HazQlqkjCE55X65ubm0a9QIqFshZe/mGl2la5+9RUrGE3qG2wopSargVLWQkq5956a1PqK13mz7Ph9IB1oYG5WoD+cueXZ16XZ0+vRpRwFl35dMOOE2rt7AEG5WtZBqnZAAwP79+13eR1FREZG2nLKz55i0SBlGcsqNrFYrp0+fpm09CqmsrCwAmsfGVnq8aYMGFBUVSXdxD3Fni5QkVRCSrn11p5RKAXoB66p5+gKl1M9KqS+VUl28G5moDXux5I4WKXsBZS+o5MTnHnIDwzhVZ+2zd12qbyEVJy1ShpKcci/7cdw6Ph6TUhw7dqzW+7AXUi2qFlIxMQDSvc9D3FZISVIFp6otUqGyIK9LlFINgE+Bh7TWVa+WNwOttdY9gJeBBefYz11KqY1KqY11HZwq6sfeIlVpsok6XOQ5t0g1sN2ckELK/c51A0Pyyf2qtki1jIsDYN++fS7vo6ioiAin/IK6jUMUnlFTTkk+uc5+rkiIjKRRVFSdWqSys7MBaGHLMbsmtkJKJpzwDI+MkZKkCh4FBQWV+uPaW6QsFotRIfk8pZSZiiJqptZ6ftXntdantdZnbN8vAcxKqcTq9qW1fktr3Vdr3TcpKcmjcYvqnatrX61bpGwXh9Ii5RnnuYEh+eQBVVukosLCSIqOrneLVKy0SPmEc+WU5JPr7OeK+MhIEqOj69e1z1Y42TW1LcorLVKe4fZCSpIquFRtkUqMigLg6NGjRoXk05RSCngXSNda/6+GbZratkMp1Z+KPM3xXpSiNs412URdW6SkkHK/893AEJ5RtZCCiu5L9S2kTCEhxMbGSouUgSSn3Md+roiLiKhzi1RWVhaNo6MJc1omAH7r2ictUp4Rev5NXCdJFXwKCwuJsg2OBGjcoAGRoaFkZmYaGJVPuxC4CdimlNpie+wvQCsArfUbwCTgHqVUGVAEXKdluh2f5Rgj5dQiZTaZiI6Orn2LVJMmwG+FlEw24R6u3MAQ7me1WikpKanUtQ8qxkn97GLXPq01GWvXOroEOouPj5cWKYNITrmXo0UqIoLE6Gh21LGQqtqtDyrOJ1FRUdIi5SFuK6QkqYJTQUEB0U2bOn5WStE6IaFW/d+DidZ6FaDOs80rwCveiUjUl71FKsapkIKKi7zariMV06oV8FtRJi1SblPtDQxb11nhIfaFdKu2SLWKj+fz9euxWq2Yqqx5U9Xq1avZk5PDQ4MHn/VcQkKCtEgZR3LKjap27Tu2e3et95GVlUXbKhNNQMV1WfPmzaVFykPc2SIlSRWEqs7aBxXT20qLlAgWeXl5xMbGVlrjBiou8ly9W261Wjlz5oyje2ADKaTcypUbGML97OvaVNciVVpaypEjR0hOTj7nPt5++21iwsK4osvZk5dKi5RxJKfcy7lrX8u4OHJzc8nNzSXeNsulK7KzsxnaufrJsps1ayYtUh7itkJKkiqwlZeXc/DgQfbv38+RI0fIy8tjwYIFlJSUnLVmQev4eNakp6O1xjbU55yOHj3K1KlTeTIlhR4vvOChdyCEZ+Tl5RFXTXeK2twtt3fhs7dqmU0mIiIipJASfq2wsBA4u5ByXkuqaiFVVFTEsWPHyM3N5eTJk8ydO5frunU7a5FRqMixvXv3eih6Ibzn1KlTmM1mosxm2idWzC2VkZHBgAEDXPr9vLw88vPzSa6mRQqgefPmbN682W3xit+4dYyUCBz5+fl89913fPvtt6xdu5aff/7ZcVK0axgZyT8vuYRb+/Sp9HhKQgL5+fmcPHmSRk7jp2ryzTffsHDhQk63acOK5593qfgSwlfUVEjFx8c7ZlE6H3vB5DzOKjY2Vgop4dfs54zquvYBTJ8+nZUrV7Jjxw52795NZmYmJ0+ePGs/U6qcY+xq0+orhC87+vXXxIeFoZQizVZI7dq1y+VCqqbFeO1aHTnC7F9+4aqrruLWW2+lf//+NG7c2D3BBzkppIRDbm4u8+fPZ86cOXy7fDkWq5WI0FB6NW/OlO7d6ZiUREpCAk1jYogND6dRVNRZs8PAb3cbMzMzXSqkMjIyAPg2M5OZM2dy4403uveNCeFBubm5NbZIbd261aV9VDeFuhRSwt/V1CLVMi6OmLAw3n77bQCS4+JIS0zkijZtaN6zJ0nR0cRFRBAXEUGz2Fg61DDLb23HIQrhq/KKix2zvbaOj8ccEsKuXbtc/n3HYrzVnIsAHho8GFNICK8tX878+RVzwbVv355hw4YxcuRIRo4cicymXTdSSAU5rTXffPMNb7/9tqOrXkpCAnf1788l7dszoGVLIqqcBM8nxVZI7du3j379+p13+4yMDFq3bk2j0lKeeuopKaSEX8nLy6Op04QrdpEZGZw4fJgTJ06QmFjtMmAOmzZtAiqfBGNjY2XWPuHXaiqkIsxmfn7oIQosFhIiIx1jAmsrISGBM2fOUFZWRmg1N/WE8Be5xcWORaZDTSbaNGzouMnsCsdivDW0SEWHhfHo8OE8MGgQW44cYUNWFj8ePMicjz7inXfeQSlFnz59GDduHOPHj6dnz57SO8hF8pcnSOXn5/Phhx/yyiuvkJGRQUJkJFN69OC6Hj3o1bx5vRKota3bhqsTTuzatYtOnTrRpaCAF3/8UU6Kwq/k5eXRoUOHsx4fkpLC2+vX06ZNGyZNmsSFF17IsGHDaNu27Vn59fJf/0rHpCT6tGjheExapIS/sxdS1Y1vahgVRUPbuoN1ZR+In5ube96bFUL4smNnzlQqgtolJta6RSokJMSxZlRNosLCGNS6NYNat+ZBwFpezpbDh/lm716W/fILT/7jH/zjH/+gdevWXHXVVUyaNImBAwdKUXUOcrUaZA4fPsxLL73EG2+8QV5eHn1btOCNK65gYufOtW55qkmD8HCSkpJcKqTKy8vZvWMHF0RE0CopibKyMg4fPkwr2zTQQvi6vLy8amdWGtepE2vuuYf/rlzJwlmzeP/99wFo1aoVo0ePZuzYsVx88cXs3LmTzYcP8+zYsZVOVrGxsRw8eNBbb0MIt6upRcpdwr/9FpBCSvg3rTUHc3MZ1Lq147G0xESWrVt33hvLFouF2bNn8/bzz9MqLg7zeZYTqMoUEkKf5GT6JCfzx2HDOFFQwJcZGXyRns4rL77I//73P1q1asW1117L9ddfLy1V1ZBCKkjs3r2bZ599lunvv09ZeTkTOnfmd5Mm0a9lS4+8Xps2bVxaS+rQoUMUlpbSvlEjxwDk/fv3SyElfF5paSn/93//x4kTJ2i4axfYFtN11rFxY96ZNAmtNbtPnGDV/v18u3cvsz78kLfeeovw8HASExNpEBbGtd27V/rdmJgYaZESfq2mySbcxT6mJD09nXbt2nnkNYTwtJMnT3LaNqzCrn1iIqWlpezbt4/27dtX2r60tJQVK1Ywe/ZsFixYQG5uLj2bNeN/48bVO5bE6Ghu6t2bm3r3Jq+4mC8zMpi/fTvPP/88zz77LJ06deKGG25g8uTJpKamurTP4uJidu7cyaFDh7joooto0KBBveP0JVJIBbjNmzczbdo05s2bR7jJxE29enH/oEGkNGzo0ddtWVDAmp07a5zRzM7edN0uMdHRrL1v3z6GDh3q0fiEqIuysjK+/fZb5s6dy2effcaJEye4tnt3pp5nZiWlFB2SkuiQlMTt/fphKSvjx4MH+Wr3blbu28ctF17oWEPKLiIjg7xff/Xk2xHCozzdInVB69a0adiQm266iWXLltG/f3+PvI4QnmTvvVOpkLJN1LVr1y7at2+P1Wpl5cqVzJo1i3nz5nHy5Eliw8MZ27EjV4wdyyXt2hFSZS3D+oqLiOA625CPk4WFLNy5kzlbt/L444/z+OOPM2jQIK6//nquuOIKWjh1S7c7fPgwr776Kq+99ppjds34+HgeffRR/vSnPwVMy5YUUgFIa823337LM888w7Jly4gND+f3F17I3QMH0thLdwJGtmvHvO3bSU1N5fe//z333nsvDasp3uyDKdMSE0mIjEQpxf79+70SoxCuKC8vZ9WqVXzyySfMmzePEydO0KBBA0alpHD9qFFcXOVuoSvCQkMZ1qYNw9q0qXGbmPBwzpSU1Cd0IQxV04K87hIXEcGiKVMY+/77DBwwgNFjxjB16lQuu+wyGWcr/Ea1hZStq+o777zDsmXL+PTTTzly5AjRZjNjOnTgiksv5eJ27Qj30nHeMCqKW/v25da+fTmYm8un27Yxd9s27r//fu6//366d+/OhRdeSMuWLSkqKmLz5s0sXbqU8vJyxnXsyFWXXEJseDhvrl/Po48+Snl5OY899phXYvc0+UsTQMrKypg3bx7//e9/2bRpE42jo/n7yJHc1q+fowuEt1zfsyedGjfm3999xxNPPMEzTz3FbXffzUMPPUQbp4vHjIwMYsLCaNKgAUopmjdvLoWUMJzWmp9//pmPP/6YWbNmkZWVRWRoaMUJbORILm7XzmMXh3Yx4eGUWK2UlJQQXsdZzYQw0t69ewlRymNd+6Bi6vTld9zBOxs2MP2HH5j45Zc0bdqUKVOmcNttt5GWluax1xbCHeyFVGunQio+MpKOSUksWrSIcJOJi9u3Z9KFFzIqLY2oaiZv8aZW8fH8fsgQfj9kCLuPH+fz9HRW7tvHzPfe43RJCSFKkZKQwL0DBnBLnz60cVoGZ3ibNty3Zw9/+ctfSElJ4frrrzfwnbiHFFIBICcnh3feeYdXX32VrKws2jVqxIuXX8613bu7bQKJuujZvDmzb7iBnUeP8tKaNbzx6qu8+uqrTJgwgfvvv5/Bgwezbds22icmOpp4U1JSpJAShtm9ezezZs1i1qxZpKenExoaysjUVJ648krGdOhQ52ma6yLG9lr5+flSSAm/s2/fPt555x2u79Gj1gPga6txgwb85aKL+NPQoSz75Remb97Mf//zH5555hkGDRrEzTffzDXXXEOC04WqEL5i7969NI6OPmt2yx/uvpuSsjKizGa3d9tzl7SkJB5JSuIR23CMQosFU0hIjS1lISEhvNCmDZktW3LPPfcwePBgWnporL63SCHlp7TWrF69mrfffpvZs2dTUlLCkJQU/nP99Yxq396nkq5zkya8ccUVPDFyJO9u2MD7S5fy2WefYTabKS0t5aZevRzbpqSksHr1agOjFcHml19+Yd68ecyZM4ctW7aglGJQq1Y8P24cEzp3rvcUzXXVxNYNd/v27QwfPtyQGISoi7KyMv74xz8SGhrK30aM8NrrhppMjO3YkbEdO/Jrfj5ztm5l5k8/cffdd3P/737HmHHjuPbaaxk3bhyxNay3I4S3ZWZmklrN0AezyeTxmxDu5kprWVhoKK9fcQWD332X2267ja+++sqnrllrSwopP7Nv3z4+/vhjpk+fzu7du4kJC+OmHj24rW9fOlcza5gvaR4by+MjR/KHoUP5bMcOth45Qv+WLbnUaYxJSkoKs2bNkrWkhMeUl5ezadMmFi1axIIFC9i+fTsAfVu04P9GjWJily4094GLrEvatyc+IoLXX39dCinh8/Ly8li+fDmff/45X3zxBTk5OTw6fDjNDMqlpjExPHDhhdw/aBA/HznC3G3b+Ozbbyu6SoWHc/HFFzNx4kQuu+wymjVrZkiMQkBFITUgyFpLUxs25OmLLuKhL77gxRdf5Pe//73RIdWZXKn6OK01GRkZLFy4kPnz57N+/XoABrVuzf3jx3NFly5e7W7kDpFmMzf07MkNPXue9VxKSgpWq5VDhw7R2mlNBSHq4+jRo6xYsYJly5axdOlSjh49SohSDGzVin+PHs3lnTqRfI7ZJY0QFRbG5F69eHP+fI4cOSIXe8KnWCwW1q9fzzfffMPXX3/Njz/+iNVqJT4igkvbt2fcyJGM69jR6DBRStGzeXN6Nm/OU5dcwvrsbBbu3MkXq1axePFiAHr16sWoUaO4+OKLGTRoEJGRkQZHLYKFxWIhKyuLa1NSjA7F66b06cOyX37hz3/+M8OHD6eXU+8kfyKFlA86duwYK1eudFz42Qci9mzWjCcvvpiJXbpUGpQYSJJWrgQqWt6kkBJ1obUmMzOTH3/8kVWrVvHDDz+wc+dOABIiIxnZti2XDBrEJe3bG9Ztz1W39+3Lqz/+yKOPPsof//hHOnfu7NddIIT/Onr0KBs2bGDt2rWsXr2adevWUVRUhKJiPOxDgwZxcbt29EtOJtRHuyOFhIQwsFUrBrZqxf+NGsWOo0f5avdulu/Zw7PPPMO0adMICwujb9++DBw4kD59+tC1a1dSU1OJiYkxOnwRgA4ePEh5eXmlGfuChVKKl8ePZ/AbbzBhwgRWrlxJih8WlG4rpJRSo4EXARPwjtZ6mrv2HchOnDjB9u3b2bJlCxs3bmTdunXs2bMHqFiQ88Jmzbj3ssu4tH17WtoWrA1k9gJx3759Adud6Xy5oipm3ngRGAsUArdorTd7PVA/cObMGTIyMtixYwfbtm1jy5YtbNq0iVOnTgEQExbGgFatuPbiixmamkr3Zs0w+VEh0qZRI27o2ZPp06czffp0GjRoQO/evenRowfdunWjS5cudOjQgUZOsyKJ6sk5yjVnzpxh9+7dpKenO/Lqp59+4tChQwCYlKJr06bc3L07g1NSGNy6NQk+fkOiOsr2Pro2bcojQ4eSX1LCmgMHWHPgAD8eOMCr69ZRYrU6tm/QoAExMTGEh4djNpuJjo4mKSmJjh07MmjQIC655JKgy0PJqfpLT08HCMpCCqBRdDRzJk9m/OzZjBgxgkWLFtG1a1ejw6oVpbWu/06UMgG7gUuAbGADcL3WemdNv9O3b1+9cePGer+2LystLSUnJ4ejR49y5MgRDh06xMGDB9m3bx979+5l9+7dnDhxwrF9s5gY+rRoQb/kZAa1bk3P5s39bqBhfVnKyujw3HNEJiTwySefcMEFF2A2cObBmiilNmmt+9bh986bK0qpscD9VBRSA4AXtdbnXvGVwMoprTWFhYUcP36cY8eO8euvv3L48GEOHTrEgQMHHDl05MgRx++Em0x0bNyYHs2a0at5c/olJ9OpcWO/KpxqcuDUKVYfOMBPhw/z8+HD7Dh6lILSUsfzDRs2pG3btqSmptK6dWuSk5Np3rw5TZs2pXHjxiQlJREXF+ezrVl1zada7D/oz1Hl5eXk5uZy/Phxjh8/ztGjRzl8+DDZ2dlkZWWxf/9+9u3bx69Oi0CHhobSPiGBrk2bOvKqZ/PmZ80uFohKrVZ+OXGC9GPHyM7L49czZzhTUoLFaqWsvJwzFgtH8/P55cQJCkpLCVGKESNHMmnSJMaNG1ftAqXe5Gs5FWj55Aqr1UpeXh4nT54kJyenUu4dOXKEzMxMli5dSmRICD8/9JDP95DwpJ8OHWLSzJnkWSzcc8893HjjjfTp0weTj1wDnyuf3NUi1R/Yo7XOtL3gLGACUONJ6ly2b9/Ovn37znr8fEWf8/Naa8fPzv9W/V5rTXl5OeXl5WitsVqtZ32VlpZW+iopKaG4uJji4mKKioooLCzkzJkz5Ofnc/r0afLy8jh16hR5eXlnxaiomHShdUICl7VqRbvevenSuDFdmjShiXQdICw0lEVTpjBl2TKGDRsGVKyE3ahRI+Lj44mLiyMmJoYGDRoQFRVFZGQkkZGRhIeHEx4eTlhYGGazGbPZjMlkIjQ0FJPJhMlkIiQkBKWU499zfTVq1IghQ4Z44i26kisTgOm64mBdq5SKV0o101ofOXt353fmzBm++eabGp+vmleu5I09X5zzp7y8HKvV6vi3rKyMsrIyR95YLBYsFosjf4qKiigqKqKgoMCRP3l5eeTl5ZGbm0tJNYvRhihF89hYWsbFcVGTJrTt1Il2iYl0SkqiTcOGPtulqL5aJyTQOiHBMa6wvLycg3l57Dp2jF9ycth38iT7cnLYtHcvC/PyKt1JtwsJCSE+Pp74+HhiY2Np0KAB0dHRjvyx501YWBhhYWFEREQQGRlJVFSUI9fsOWbPKXu+AJW+r86wYcOMnCnNreeoNWvWVLoJ5qzqeajq987/lpeXO753zivnfLJ/lZWVOf51zieLxXLW+aimnKruHBpmMtE8NpZW8fFc3KwZbbp0oW2jRrRv1Ih2jRoRFqST/phNJjo3aXLeSZys5eVsPnSIpbt389nmzdy9fDkA7dq1o1evXqSmppKUlERMTAwRERGOc1JN5yGgUh7VlFepqal069bNje+41tyWU4cOHWLz5rM7XdT2mq+6x2r6qpprVfOtunOY/frP/lU156rmXX5+fo2xNwgLo1lMDHf17ctdAwYEdREF0KtFC9bfey9/X76c1199lZdffpmwsDBSU1Np3LgxCQkJREdHVzoX2c9H9uu+0NDQSl/2PLP/a/+qLu9CQkKYMGFCnWJ311/IFkCW08/ZVNxJr0QpdRdwl+3HM0qpDDe9PkAiUP2ZzXckajhx6PRpDp0+zZoDB4yOpzo+9znm5uaSm5vr/JAvxFjXAVyu5Ep127QAziqkPJxT7uDW/6tyrcnOyyM7L48fDx50126d+cKxVRsux1teXs7Jkyc5efKkh0M6p5ri9fSASF84R4EPHl8Wq5X9p06x39YdtgY+F3ctGBL7nj17HN3066i+cRueU7XMJ38+xqCW8Z+xWPglJ4dfcnJ4be1aD4blMp/7/C0WCxkZGWRkuPxn2JPvocZ8clchVd1tyLNuJWit3wLectNrVg5AqY2ebMZ2B4nRPfwhxnNwJVdcyifwbE65g7/9X0m8nmVgvIafo8D//r/s/DVu8N/Y/SDu8+ZUbfLJD97vOUn8xjPqPbirw3w24Lw0cTJw2E37FiKQuJIrkk9CuJfklBDuJTklBO4rpDYA7ZVSqUqpMOA6YJGb9i1EIHElVxYBN6sKA4G8uo6PEkIAco4Swt0kp4TATV37tNZlSqn7gK+omAbzPa31DnfsuxZ8tnuTE4nRPfwhxmrVlCtKqbttz78BLKFixr49VEx/fqtR8bqBv/1fSbyeZUi8PnKOAv/7/7Lz17jBf2P36bg9kFM+/X5dIPEbz5D34Jbpz4UQQgghhBAimPjmoiJCCCGEEEII4cOkkBJCCCGEEEKIWvLbQkopdbVSaodSqlwpVeN0h0qp/UqpbUqpLUopry6rXYsYRyulMpRSe5RSj3o5xoZKqa+VUr/Y/k2oYTuvfo7n+0xsEzG8ZHt+q1Kqt6djEufnD8d8lTh88viv5vX9Kh9ciHe4UirP9nluUUo9YUScnuZv+eAUj1/khVMcfpUfzoIxV/w1L5z5W444xeO3uQI+mi/nWvnZl7+ATkAH4Dug7zm22w8k+mqMVAzS3Au0AcKAn4HOXozxP8Cjtu8fBZ4x+nN05TOhYjKGL6lYy2IgsM6I/2P5Ouv/zueP+Sqx+NzxX5fPy5fywcV4hwNfGBWjFz8Lv8oHp5h8Pi9q8/n5Un7UIfaAyxV/zYsq8flNjtTmM/XVXKlF/F7PF79tkdJap2ut3b3qvFu5GGN/YI/WOlNrbQFmARM8H53DBOBD2/cfAhO9+No1ceUzmQBM1xXWAvFKqWbeDlRU5ifHvDNfPP6r8rd88KX/X0P5YT7Y+UNe2Plbfjjzxf97j/PjvHDmTzli58+5Aj56TPhtIVULGlimlNqklLrL6GCq0QLIcvo52/aYtzTRtjWKbP82rmE7b36OrnwmRn9uou586f/OF4//qvwtH1yN5QKl1M9KqS+VUl28E5pP8qX/Ozt/yAs7f8sPZ5IrNfPV/zM7f8oRO3/OFfDRfHHLOlKeopRaDjSt5qm/aq0XuribC7XWh5VSjYGvlVK7tNYrfShGVc1jbp2T/lwx1mI3Hv0cq3DlM/H45yaq5w/HfKUX87/jvyp/ywdXYtkMtNZan1FKjQUWAO09HZgn+Fs+OF7U//PCzt/yw1nA5oq/5kWlAAInR+z8OVfAR/PFpwsprfXFbtjHYdu/x5RSn1HRNOi2g9gNMWYDLZ1+TgYO13OflZwrRqXUUaVUM631EVvz7bEa9uHRz7EKVz4Tj39uonr+cMw788Pjvyp/y4fzxqK1Pu30/RKl1GtKqUSt9Qkvxeg2/pYPdgGQF3b+lh/OAjZX/DUvnAVQjtj5c66Aj+ZLQHftU0pFK6Vi7N8DlwLbjY3qLBuA9kqpVKVUGHAdsMiLr78ImGL7fgpw1p0iAz5HVz6TRcDNthlmBgJ59mZ24fOMPuad+eLxX5W/5cN541VKNVVKKdv3/ak4F+V4PVLf4Ev5YOcPeWHnb/nhTHKlZr6YF878KUfs/DlXwFfzpTYzU/jSF3AFFdVpCXAU+Mr2eHNgie37NlTM6vEzsIOKJmWfitH281hgNxWzkXg7xkbACuAX278NfeFzrO4zAe4G7rZ9r4BXbc9v4xwzN8qXV48nnz/mq8Trk8d/NXH6VT64EO99ts/yZ2AtMMjIeD34OfhVPjjF4xd5UYvjzafyo5axB1yu+GteVHkPfpUjtTjefDZXXIzf6/mibC8shBBCCCGEEMJFAd21TwghhBBCCCE8QQopIYQQQgghhKglKaSEEEIIIYQQopakkBJCCCGEEEKIWpJCSgghhBBCCCFqSQopIYQQQgghhKglKaSEEEIIIYQQopakkBJCCCGEEEKIWpJCSgghhBBCCCFqSQopIYQQQgghhKglKaSEEEIIIYQQopakkBJCCCGEEEKIWpJCSgghhBBCCCFqKdSoF05MTNQpKSlGvbwQ9bJp06YTWusko+NwJjkl/JXkkxDu5Ws5Jfkk/Nm58smwQiolJYWNGzca9fJC1ItS6oDRMVQlOSX8leSTEO7lazkl+ST82bnySbr2CSGEEEIIIUQtSSElhBBCCCGEELUkhZSfsFqt/P3vf+f48eNGhyJEQDt27BiPPPIIZ86cMToUIXya1pqysjKjwxBCCMMYNkZK1M7u3bv55z//SZs2bZgyZYrXX7+0tJTs7GyKi4u9/tpGioiIIDk5GbPZbHQowkvuvfde5s2bR+/evZk8ebJb9hms+VOV5FNgmTZtGk8+/jhjx4/nueeeIzU11WuvLTlVQXJKAOQ9+SQA67OyKLRY6PN//0dycrLLvy/5VKEu+SSFlJ8oKCgAwGKxGPL62dnZxMTEkJKSglLKkBi8TWtNTk4O2dnZXr1AEMb5/PPPmTdvHgArVqxwWyEVjPlTleRT4Fm6dCkx4eEsXbqU2NhYPvjgA6+9tuSU5JT4zeniYv64ZAmzt24FIGTGDD7+5BOuvfZal35f8qnu+SRd+/yEvZAqKSkx5PWLi4tp1KhRUCWYUopGjRoF/R2aYPKH226jc+PGjE5LY/ny5Wit3bLfYMyfqiSfAovVamXTpk1c0aULF110EZs3b/bq60tOSU6JClarles/+YS527bxp2HD+OKWWxjQsiWTJ09m4cKFLu1D8qnu+SSFlJ8wukUKCMoEC8b3HKzOnDnD7hMnuLJrVy5p356srCz27Nnjtv3LsSSfQSDZtWsXBQUF9G7Rgt69e7Nz506Kioq8GoMcT/IZCHj66adZfeAAr0yYwF8uuojBKSnMmTyZro0bc9fkyRQWFrq0HzmW6vYZSCHlJ4xukRIi0GVkZADQPjGR4W3aABXd+4QQZ9uwYQMAvVu0oOMvv2C1Wtlq61YkRLDKe/JJx3glb9i0aRNPPvkk13Tvzg09ezoejwkP5+lRozhWUMBbb73ltXiCkRRSfsIXWqSE9ymlTEqpn5RSXxgdS6BLT08HoENiIm0aNqRly5ZSSAUYySf3Wb9+PTExMbRv1IgezZoBFRd1QgSr7Oxsbpo9mx1Hj3rtNf/85z+TkJDAs2PHnvXc4JQUhqSk8Mwzz3i9tTiYSCHlJ+xNs9IiBY888gidO3fm/vvvd2n7oqIihg0bhtVqBSr+2M2ePRuoKEyHDh3qy1P4PgikGx1EMEhPT8ekFG0aNkQpxZAhQ1i/fr3RYbldbfOnOn6cU5JPbrJhwwb69OlDSEgIyXFxNIyM9Po4KV8R5DklbD755BM+T09n7Pvv8+OPP3r89ZYvX86KFSv461//SlxERLXb/GnYMH799Vc++eQTj8fjLv6WT24tpJRS8UqpeUqpXUqpdKXUBe7cfzCzt0id/u47YwMxWGZmJqtXr2bnzp28/PLLLv3Oe++9x5VXXonJZAIqumvZT/hhYWGMHDnSkWC+RCmVDFwGvGN0LMEgPT2dNg0bEhZaMZlpamoqhw4dCqiLl7rkT3X8Mackn9ynpKSEnzdvpkdpKVAxrqBn8+ZB2SIVzDklKvv+++9JjosjITKSW8eP9+hraa3585QpJMfFcUNOTo3bDU5JISUhgTlz5ng0Hnfxx3xyd4vUi8BSrXVHoAdy589tHF37bNV1MMrIyGDYsGEcOHCAXr16OT6T85k5cyYTJkwAYNWqVTz88MPMmzePnj17sm/fPiZOnMjMmTM9GXpdvQD8CSivaQOl1F1KqY1KqY2yWHP9pKenk5aU5Pi5ydatWK1WDh06ZGBU7lPX/KmOn+bUC0g+ucWuXbsoLS93dOkD6NGsGdu3bw+qXhOSU8LOarXyww8/cHG7dkzp3ZuMEyfIOUeBU1/ff/89mw8f5pEhQ4g4x5pHSimu6NKF5cuXezQed/DXfHLbOlJKqVhgKHALgNbaAsiAHjdxFFI+cHf8oYceYsuWLW7dZ8+ePXnhhRfOuU2HDh2YMmUKKSkp3HHHHS7t12KxkJmZSUpKCgCDBw+mX79+/Pe//6Vr165AxR9A+8BpX6GUGgcc01pvUkoNr2k7rfVbwFsAffv2dc9c3UGotLSUPXv2MHbgQMdjLePjAThw4ACtW7d222v5U/5Uxx9zSvLJvQ4cOABAa1uOAHRr2pSysjLS09Pp6TTo3Rskp4TRtmzZwunTpxmckkLTmBgA1q1bx9hqxi65w7/vuovEqCiu69HjvNtO7NKF51et4rPPPnPpOJV8qh13tki1AY4D79sG876jlIp23kDu9tWdtEhV2LZtGz1sfzgefPBBCgsL2blzJ9dccw333HMP8+bNY9OmTbz55psAnDhxgninkz1U3PXo0KGD42eTyURYWBj5+fleex8uuBAYr5TaD8wCRiilZhgbUuDas2cPZWVllVqkWsbFAb9dNAaC6vInMzOT22+/nUmTJgEVf2umTJnCnXfe6bhjFwA5JfnkRllZWQC0sOUIQGpCAgD79+83IiTDuJJTVX+GgMgp4WTpX/4CwIWtW9OreXNMSnlsnNSOHTtY9ssv3DVgAJHnaI2y6960Ke3atfOL7n2u5NOCBQu48847mTBhAsuWLQOMzSe3tUjZ9tUbuF9rvU4p9SLwKPC4fQO521d39skmfKGQOt9dBU/asWMHXbp04eTJkyiliIqK4ssvv+T+++9nyJAhjB8/nkWLFvH8888zdepUIiMjKy2ulpOTQ1xcHOYqf3xKSkqIqGGwphG01o8BjwHY7qD/QWt9o5ExBbJdu3YBkJaY6Hgs2XaR6O4LQ1/LnzZt2vDuu+86TlLz589n0qRJXH755Vx77bVMnjyZPn36+HVOST65V1ZWFuaQEBpH/3avtJVTC663+XpOVf0Z8PucEpWtPnCANg0b0iw2FoAuTZp4rJB66aWXiAgN5fa+fV3aXinFVVddxX//+1/y8/OJsbWY1cTX82nixIlMnDiRU6dO8Yc//IFLL73U0HxyZ4tUNpCttV5n+3keFYWVcANpkYL8/HzMZjNRUVFs2bKFzp07A3DTTTcxa9Ys/vjHPzr6AEdERHD06FESEhKwWq2OpNq3bx/NmzevtN+cnBySkpLOSjIRPOyFVHunQirCbKZJgwYB0yJVU/5UlZ2dTcuWLQEcA3VBckr8Jisri2axsYSE/HYJ0TAqimizOahapFzNqZpITgWOjdnZXNCqlePnfsnJrF+/3jFrnLvk5uYyY8YMru7WjUbR0ef/BZtLLrnEMY7LV9U2n/71r39x7733On42Kp/cVkhprX8FspRS9razkcBOd+0/2PnSGCmjbN++3dG/9eTJk46m28aNG/Pqq68ybdo0Em0XwgkJCfx/e/cdX1V9P3789cneey8SRtg7LAFxMZyoVQuuuqWt/rT6bW21rbXWWrW1tVOpAy3iqLgVQXGjbIEQZiAkhEBCQhZkJ5/fH7k3DSEh69x7zr33/Xw88iC599xz3vfe8+be9/msqqoqAObMmcPXX38NwLBhwygtLWXUqFF88803AHz22WcO68dsBK3151rri8yOw50dOHCA6KAgQv39T7o9NTzcbQqprvKno5SUFAoLCwFoafnfvAzuklOST/138OBBkm1X3u2UUqRGRHhUIdXTnOqKu+SUp6upqeHoiRMMjIpqu21SairV1dXs2GHs1+AlS5ZQU1PDLZMn9+pxIz/7DH9vbz799FND4zFST/NJa819993H+eefz4QJ/2uvMSufjJ61707gZaXUNmAc8HuD9++x7IVUvQe3SE2bNo3//ve/AGRmZrZ9YB84cIDbbruN66+/np/+9KcAHDp0iDTb1aE77riDF198EYCQkBDWr1/P9u3bOeOMMwBYtmwZt912m5OfjbCSwsLCU74YAm71xbCr/CkrK2PRokV89913PProo1x++eUsX76cH/7wh1x88cVtj5ecEnYHDx5s6/raXpob5UtP9DSnOv5tJznlHuxjBtvnxOSUFADWrl1r2HFaWlr4xz/+wbRp006aMbMnAn19mZyaaulCqqf59Le//Y1PPvmEN954g6effrrt8Wblk5FjpNBabwF61mlT9Ip07TvZ6NGj+ec//wlAeno6ixcvbrvv+PHjhIWF4W9rXRg/fjxnn302zc3NJ3VVgtbZXS699NKTBiEKz9PVF8PU8HA+2LiRlpaWk7oxubr2+RMdHX3ShxHACy+8cNLfklPCrqWlhUOHDnFpJ1fE0yIi2JCba0JU5usupzr+7eo5pZRKBV4CEmhdUmCx1vopc6MyR6eTr0RFERISwrZt2ww7zieffEJubi6/+c1voA95dmZGBr///HPKysqIjo42LC5H6C6f/t//+38n/W1mPrnPNwM31zbZhAd37WtPKcU111zT9rq0V1RU1NYyZXfTTTedkkzQujDb9ddf77A4hWs4ePAgKZ20SKVFRNDQ0EBxcbEJUTnO6fKnM5JTwq64uJjGxsaTvjTapUVEUF5eTmVlpQmRmcsDc6oJuFdrPRyYCvxYKdW7QWJuoqCgADi5RUopxbDwcLa+/75hx3n66aeJiYk5adKS3pg1cCBaaz740Y8Mi8lRXCmfDG2REo4jLVKnmjlzZqe3Z2ZmOjkS4cqOHz9ORUUFSV107YPW7qOJvexKYXVd5U9nJKeEXdvV9y4uPEDrzH1jxoxxZliW4Ek5pbU+DBy2/V6tlNoJJOOBY+MPHjyIApI6zIY3PC6OFbt3G3KMQ4cO8e6773LH1KnU/eEPfdrH+KQkQvz8+PrAASxXlnfCVfJJWqRchBRSQjiGfWKFzq6wu+NaUkL0R2fdmOzS2l14EJ5DKZUOjAfWdbOpWyooKCA+JAQ/n5PbJobFxVFaU0NJSUm/j/Hvf/+blpYWbpw4sc/78PX2ZnxSEhttn3nCGFJIuQgppIRwjNNdYbe3SNm7bgjh6doG1neWL3LhweMopUKA5cDdWuuqDvfdppTaqJTaePToUXMCdIKCgoJOx9iOiIsDWtdG6o/SX/+axYsXM3fuXNLbzQzYF5NSUsg+coTa2tp+7Uf8jxRSLqD9fPgyRkoIY7W1SHXyxTDU35/g4GAOHz7s7LCEsKTc118nwMeHqKCgU+6LCQ4mMDBQWqQ8hFLKl9Yi6mWt9Zsd79daL9ZaZ2mts2JjY50foJMcPHiw0xbaYbbnvH379n7t/4Nduzh8+DA3REb2az8AWSkpNLW08N133/V7X6KVFFIuoP2VAzOnP9dam3Zss3jic/Y09ivsiZ0UUgDx/v4UfPxxv48j55K8Bu7gUGUlKeHhKKVOuU8pRXp6utMKKTmfzHsNVOsJ8BywU2v9pClBWIDWussWqYTQUCICAvrdIvXshg2khocze8iQfu0HWgsp6Hxadsmnvr0GUki5AHu3Ph8vL9O69gUEBFBWVuZRiaa1pqysjICAALNDEQ508OBB4uPj8ffpfO6d+JAQio8f79cxPDF/OpJ8cg9FVVWdTsxiN2DAAKcUUpJTpufUdOA64Byl1Bbbj8etGHzs2DFqa2s77dGglGJ4XFy/WqRycnL4+sABbp40CW8DluCICwkhLSLilEJK8qnv+SSz9rkAeyEVERDAiYYGU2JISUmhsLAQd+7n3JmAgABSbFdwhHsqLCw87XscHxLC9n5Of+6p+dOR5JPrO3L8OFNti152JuHoUTbu2uXwOCSnWpmVU1rrr4FTmyU9TGdTn7c3PC6ON3Ny0Fp32orbnb/97W/4e3tz7fjx/Yqzvazk5FMKKcmnVn3JJymkXIC9kIoMDKTcpAGCvr6+ZGRkmHJsIRzp4MGDDDlNl4n40FBW93ORUckf4Q601pQcP058SEiX2ySHhVFaU0NdXZ1DW0okp4QVtE2+0kUhNSIujuc3buTgwYOkneYCRGeOHTvGSy+9xFVjxhATHNzvWO2yUlJ4c+VKioqKSEpKAiSf+kO69rmAthapwECataZZZu4TwjCFhYWkpqZ2eX9CSAjVDQ1teSiEp6qqqqKuqYm40xVSti+UhTLFsvAA3bVIjbGtP9iXyR3+/v3vU1tby6KpU/seYCcmJCcDsGnTJkP366mkkHIB9pWdI2xX9xpM6t4nhLuprq6msrKS2NN0RbJ/aTxy5IizwhLCkuw50F2LFPzvSr0Q7qywsBBfLy9iOpnFEmBkfDxeXl5s3ry5V/ttbGzk3+vXMzM9nZHx8UaE2mZUfDwKeh2T6JwUUi6gfdc+gPr6ejPDEcJt2K+an27wfIJttXoppISn60khZb8yL4WU8ATFxcXEhYTg1cVEEMF+fgwbNqzXLVKvvvoqhVVV3HHGGUaEeZIQf3+GxMTIFOgGkULKBdgLqXAppIQwVE8KKfuXRllLSng6eyF1uq59SdIiJTxIcXExsd2MX5owYUKvWn+01jz++OOMiItjjgFTnndmTEKCtEgZRAopF9B+1j6Qrn1CGEUKKSF6rictUoG+vkQHBUkhJTyCvUXqdMaPH8+hQ4coKSnp0T5XrFjB9u3bufOMM/o0019PjE1M5ODBgx4/S58RpJByAe0nmwBpkfIESqkApdR6pdRWpVSOUuohs2NyR4cOHQIg0dZ9rzPRQUH4eHlJ1z4XJvlkjCNHjuDr5dV2Ua8ryWFhUkgJj1BcXNztjHqZ27YBPZtwQmvNww8/TEpYGFeMHm1IjJ3pzyQY4mRSSLkA+2QT9jFS0iLlEeqBc7TWY4FxwDyllLFT9wgKCwuJCQrqcjFeAC8vL+KCg6VFyrVJPhngyJEjpx0PYpccHi6FlHB7WmtKSkqI66aQGp2QAPRscodVq1axdu1a7j3zTHy9vQ2JszNjbYWUdO/rPymkXMCJEydQShHm7w9Ii5Qn0K2O2/70tf147pLjDlJYWHjabn12cSEhUki5MMknY9gLqe6kSIuU8AAVFRU0NjYS201ORAQGMnjw4FMWwe1Ia81vfvMb0tLSuGbcOAMj7Tym9PR0aZEygBRSLuDEiRMEBwe3XTWXFinPoJTyVkptAUqAj7XW6zrZ5jal1Eal1Ebp69x7hYWFJPagkIoPDZWufS5O8qn/Dh8+3KNCKjk8nIqKCo4fP97ttkK4quLiYoBuJ5sAOCsyktUrVlBXV9flNu+99x5r167lnjFj8DtNLwmjjAoIYOMnnzj8OO5OCikXUP7llwRpjZ+tmVdapDyD1rpZaz0OSAEmK6VGdbLNYq11ltY6KzY21ukxurpDhw61rXtzOgnSIuXyJJ/678iRI6edaMJO1pISnsA+eUR3XfsA5mZmcqKxkS+++KLT+xsbG/npT39KZkwM144fb2icXRmdkEDesWOy2Hw/SSHlAmoaGgj09ZVCykNprSuAz4F55kbiXurq6igtLe1x176SkhKampqcEJlwJMmnvmlububo0aM9bpECKCgocHRYQpimrUWqBzkxMz2dIF9f3n///U7v//P8+ezZs4ffzp6NjwPHRrU3KiEBDWRnZzvleO5KCikXUNPYSLCfX1shJV373J9SKlYpFWH7PRA4D9hlalBupicz9tklhIa2DSwWrkfyqf+OHj1KS0tLj1qkUqSQEh6gN137Anx9mTVwIO+//z5anzw88+DBgzzy6afMyshgbmamQ2LtzKj4eAC2bt3qtGO6IymkXEBNYyNBvr5tfWalRcojJAKfKaW2ARtoHdPR+aUs0Sf2NaR60rVP1pJyeZJP/dSTxXjtkkJD8fb25sCBAw6OSgjzFBcX4+XlRXRQUI+2n5eZyYEDB06aKU9rzS233EJzSwtPXXyxw9aN6kxaRARh/v5SSPWTFFIuoLaxkUBfX/ylRcpjaK23aa3Ha63HaK1Haa1/a3ZM7sbeItWTrn0JtlYrKaRck+RT//VkMV47H29vUkJD2fPuu44OSwjTlJSUEBMTg3c3ywHYXTx8ONHR0SxatIjGxka01vz6179m1apVPDR7NulRUQ6O+GRKKUbGx0sh1U+GTguilPIGNgKHtNYXGblvT1bX1ERMUJCMkRLCQPYWqR7N2ictUsLD9aaQAhgQGUl+RYUDIxLCXMXFxcTFxfV4+6igIJ4891x+8Prr/OAHP8Db25ulS5dyzbhx3JSV5cBIuzYqIYFXtm2jpaWl2/XhROeMftXuAnYavE+PV9/UhJ+PT1vXPmmREqL/9i1fTpi/P6G29dlORwop4ens535PuvYBDIiIIL+83JEhCWGq4uJi4m3jjHpq/ogR3JyVxSuvvMLLS5fyw6lT+dsll5hWxIyOj+f48ePk5eWZcnx3YNg7p5RKAS4EnjVqn6JVfVMTAT4+0iIlhIGKqqp61K0PwM/Hh+igICmkhMc6fPgwoX5+BPv59Wj7AZGRlJw4QU1NjYMjE8IcJSUlvS6kAP500UXU1NRQ+utf8+i8eaa2BI1KSABkwon+MPLd+wvwM6DFwH0KoL65GT9vbymkhDBQYWVljwspaG2VkkJKeKrDhw8T34MZLu0GREQAyIQTwm31pUXKruHxx3s8tsqRhsfF4eXlJYVUPxjyLiqlLgJKtNabutlOVo3vA3uLlL907RPCMPkVFW1f9noiITRUCinhsQ4fPtzj8VHQ2iIFsH//fkeFJIRpTpw4wYkTJwjbts3sUPol0NeXwVFRbHPx52Emo8rh6cAlSqkDwKvAOUqppR03klXj+8Y+RsrXdvVCWqSE6J+qqirKa2vbvuz1hLRICU92+PDhttkreyLdllsy9kK4o94sxmt1MnNf/xhSSGmtf6G1TtFapwMLgE+11tcasW/xvxYpLy8vfL28pEVKWFpDQwN1dXVmh3Fa9u5GvW2ROnLkCC0t0ntZeBatda9bpGKDgwny9ZVCSrgl++LscT1YjNfqRsXHk5eXR1VVldmhuCTzO2iK09Jat42RAvDz9pYWKWFpt99+O9/73vfMDuO07F/uetsi1djYSFlZmaPCEsKSqqurqa2t7VWLlFKKtIgIKaSEW2prkXKDQmqkbZxXdna2yZG4JsMLKa3157KGlHHsrU8BtvFRfj4+0iIlLG337t3s2rXL7DBOy/7lLq0XLVKJsiiv8FD2c743LVLQ2uIrhZRwR/YWKXfo2mefuU/GSfWNtEhZnL31yb6GlL+0SAmLO3bsGKWlpWaHcVoHDhwg2NeX6KCgHj8mXgop4aHs53xvWqQA0iIj2b9/P1prR4QlhGnsLVIxvfgMsarksDAiIiJknFQfSSFlcfaxJvYWKV8ppITFlZeXU1VVZemW07y8PAZERqKU6vFjEmRRXuGh+toiNSgqiurq6rYvnUK4i5KSEsL8/Qnw9TU7lH5TSjFmzBhpkeojKaQszl40+dvGSPlL1z5hYVprysvLASw9ligvL4/UXnTrA2mREp6rry1Sw+PiABl7IdxPcXExcW7Qrc9u7NixbNu2TSZT6gMppCzO3iJlX0NKJpsQVlZTU0NjYyOAZbv3aa05sHt3r2bsg9b1NsLDw6WQEh7n8OHD+Pv7Ex4Q0KvHySB24a5KSkrcYqIJuyH5+Zw4cULWfesDKaQsrq1Fql0hJS1SwqrsrVFg3ULq2LFjVDc09GrGPrsEX1/yP/nEAVEJYV2HDx8mMTGxV11hAWKCg4kLDmb79u0OikwIcxQXF7tVITXGNuHEli1bzA3EBUkhZXEdW6T8fXykRUpYlisUUm1Tn/eyRQogOTycQ7LWhvAw9kKqL4bHxUmLlHA77tYiNSw2Fm+lpJDqAymkLK6tkLKNkfKVFimPoJRKVUp9ppTaqZTKUUrdZXZMPXHs2LG2361eSPVm6nO7lPBwDlZUGBuQcDhXzSerOLRtGzF9PO9HxMeTk5NDc3OzsUEJYRL7eoLuMPW5XYCvL5kxMVJI9YEUUhbXsWufTH/uMZqAe7XWw4GpwI+VUiNMjqlbrtAilZOTgwIGR0f3+rGp4eGUnDjRdoFDuAyXzCerOHL8eNuslb01Ii6O2tpaWU9KuI2jR48C7rEYb3ujExKkkOoDKaQsrmPXPpn+3DNorQ9rrTfbfq8GdgLJ5kbVPVcopLKzsxkYFUWQn1+vH2uf6e/gwYMGRyUcyVXzyQpqa2uprKtrm7Wyt0bIzH3CzbQtxuuGhdShQ4cs+9ltVVJIWdwpLVIy/bnHUUqlA+OBdSaH0i17IRUdFGTZ/4y3bdvWNptYb6WEhwOQn59vZEjCiVwpn6ygoKAA+N+531vD4uJQSkkhJdyGfV00d5r+HFoLKUAW5u0lKaQsruOCvNK1z7MopUKA5cDdWutTZjlQSt2mlNqolNpo725gpvLycry8vBgQEWHJQurEiRPs27ePEX0spFJtXybtXy6Fa3G1fLIC+7me2sdCKtjPjyFDhrBhwwYjwxLCNO7aIjXKVkh9+/jjJkfiWqSQsjh70eQnk014HKWUL61f+l7WWr/Z2TZa68Va6yytdVZsbKxzA+xEeXk5ERERxAYHW7KQ2rFjB1rrtu5GvZUUFoaXUtIi5YJcMZ+swH6u97VFCmB6aChfrFrVtsacEK7M3iLlboVUTHAwiaGhZMtaib0ihZTFndIiJdOfewTVumDLc8BOrfWTZsfTU8eOHSMyMpIoi3bts3cv6mvXPl9vbxJDQ6WQcjGumk9WUFBQgJdSJIWF9XkfZw0aRHVDA+vWSW9Kd6GUel4pVaKU8rhFwkpKSvD39ibU39/sUAw3JjGRLVJI9YoUUhbXcYyUtEh5jOnAdcA5Sqkttp8LzA6qO+Xl5URGRlp2jFR2djaBgYGk92ExXrvU8HDp2ud6XDKfrCA/P5/E0FB8bb0i+mJWRgZeSvHxxx8bGJkw2RJgntlBmKG4uJjYkJBeL1DtCsYlJrK3tJTq6mqzQ3EZUkhZ3CkL8soYKY+gtf5aa6201mO01uNsPx+aHVd3SrOzCSsvJzooiJqaGmpqaswO6STfvfMOwyIj8fbq+399KeHh0iLlYlw1n6ygoKCgz+Oj7CICAxmflCSFlBvRWn8JHOt2QzdUUlJCnJt167Mbn5SEBpkGvRekkLK4thYp29XAYD8/amtraWpqMjMsITpVUVdHREAA0UFBAJSVlZkc0f9orckpLu7z+Ci71IgIDh48SEtLi0GRCWFd+fn5/RofZXf2wIGsX7+eyspKA6ISrsBdJ28pKipyuxn77MYlJQGwadMmkyNxHVJIWVxdXR2+Xl542a6gRwQGAlDRx1XmhXCkitpaIgIDibIVUlbq3rdz505Ka2qYlJLSr/2khIfT2NjIkSNHDIpMCGtqbm6msLCwbf20/pg9ZAjNzc28/vrr/Q9MuAR3nbzl0KFD/RozaGUJoaEkhITw7QsvmB2Ky5BCyuLq6+vbuvUBRNoKqWPHrNGiLi1jwk5rfUqLlJUKqVWrVgFwzuDB/dpPqqwlJTzEkSNHaGxsNKRFanJqKuOTknjiiSdobm42IDohnK+uro6ysjIS+7hAtSsYm5TE1qIis8NwGVJIWVxdXd1JhVREQADwv4VPzXTw4EFSUlK49957zQ5FWMDx48dpamkhIjDQsoXUkOho0vp5dd0+UUVubq4BUQlhXf1dQ6o9pRR3z5jB3r17eXnhwn7vTwgzFNkKjEQ3bZGC1gkn9pSWcvz4cbNDcQlSSFlcfX192/gosE6LlNaaW2+9leLiYp588kmee+45U+MR5rMX91Zskaqvr+fzzz/n7EGD+r2vjKgofL28yMnJMSAyIazLXkgZ0SIFcNGwYQyKiuLRzz+XWcFcnFLqFeBbYKhSqlApdbPZMTnDoUOHAEhy4xapcbYJJ7777juzQ3EJUkhZ3CktUrZCyuwWqRdffJGVK1fy6Lx5zJ49mx/96Edtq30Lz9RWSAUGEhEYiFLKMpNNrFmzhtraWs4xoJDy9fZmSEyMFFLC7RmxGG973l5ePH7BBewtLeWqq66SBXpdmNZ6odY6UWvtq7VO0Vp7xNVUeyHlzi1SWcnJALLuWw9JIWVxVh0j9Zf772d0QgK3T57M/enpNDQ08Omnn5oakzBX+0LK28uLiIgIyxRSH3zwAT4+PkxPTzdkf8NiY6WQEm4vPz+f8IAAwmxdyo1w7uDB/Pmii/joo484IyNDZgcTLqWtRcqNC6nYkBDSIyP59ttvzQ7FJUghZXEdW6TCbR9oZhZSeXl5bD18mCtHj8bLy4sxiYmE+ftLIeXh2nftA4iOjrZEIVVdXc1zzz3HpZdeathK9MPi4sjLy+PEiROG7E8IK9qzZw8Do6IM3+/1Eyey+PLLya+oICsriylTpvD444+zadMmmYhCWFphYSHBwcGEGfRZYlWTU1L49ttv0VqbHYrlSSFlcR3HSPl6exPq52dq174333wTgEuGDwdau2vMSE+XQsrD2afkD2tXSFlhjNSzzz5LZWUliwyYwtlumG0q3507dxq2TyGsZseOHW3nutGuGjOGDXfcwcOzZ9PQ0MB9991HVlYWkUFBnDNoEA899BCrV6+23KLewrMdOnSI5ORklFJmh+JQWSkpHD58mIMHD5odiuUZVkgppVKVUp8ppXYqpXKUUncZtW9P1rFFClq7TpnZIvXGG28wJiGB9HZXKs/MyGDfvn0yJbQHq6qqAiDcdqUuJibG9BapxsZGnnzoIaalpZHVz/Wj2htuW9RXuvcJd1VRUUFRUZHDCilo/Sy7c/p0Pr/0Unbdey8vv/wy3x87lpITJ3joN7/hvPPOIyI0lLPPPpvHH3+cPXv2UPnQQ1Q+9JDDYhLidOyFlLubnJoKIN37esDIFqkm4F6t9XBgKvBjpdQIA/fvkTqOkYLWDx+zWqQKCwtZu3Ytl4w4+a2dmZEBwGeffWZGWMICKisrAdq6z1mha9+vf/1rCisr+cmMGYbuNyMyEj8/PymkhNuyn9vDbBcNHC0hNJQL9+7ljxdeyJof/pADP/85r199NYumTuXozp3cd999DB06lLMXL+a5DRukW60whacUUiPj4wkMDGTt2rVmh2J5hhVSWuvDWuvNtt+rgZ2A+59tDtZZi1SkiS1Sb731FvC/bn12w2NjiQkKku59HqyyspJgX198bF1RzS6kPvjgA/7whz/wgwkTmJOZaei+fby9GTZsmBRSwm3t2LEDgKEObJE6nfCAAOZkZvLwnDms+eEPyb77bh6ZO5fG5mbu/eAD0tLSeOSRR6Trn3CalpYWioqKPKKQ8vX2ZtKkSdIi1QMOGSOllEoHxgPrOtx+m1Jqo1Jq49GjRx1xaLfTcYwUtA7mN6tF6r9PPsmw2FgyO3y4enl5MTk1lfXr15sSlzBfZWXlSbN7RUdHc/z4cRoaGpwey7vvvsuVl17KqPh4/nD++Q45xsiRI8nOznbIvoUwW05ODkFBQaQZNPV5f6VGRPDjadP4+oc/ZOVNNzF9+nR++ctfkpmYyPvvv292eMIDlJaW0tjY6BGFFEBWSwubNmyQhXm7YXghpZQKAZYDd2utq9rfp7VerLXO0lpnxZp0lcvVWKlFqrS0lDX5+Vw0bFin949JTGTPnj2SdB6qsrLypJmMgmxXspzZKtXU1MTvf/97LrvsMobFxfHmddcR6OvrkGONq6jg4MGD7N+/3yH7F8JMOTk5jBgxAi8v681JNSUtjf9MnMiHN95IeEAAF198Mbfeeit1dXVmhybcmH3q84ivvzY5EueYmZFBU0sLa9asMTsUSzP0f0illC+tRdTLWus3jdy3pzrdGClnT0v57rvv0qI1F3Xo1mc3JiEBrTXbtm1zalzCGk5pkQoKAnDazH07duxgekYGDzzwAJcOH877N9xAXEiIw4531sCBAKxevdphxxDCLDt27GDECGsPcz5jwAA+v+02fjJjBs8++yxnnnlm25ddIYzmCWtItTc5NRVfLy8+//xzs0OxNCNn7VPAc8BOrfWTRu3X03XVIlVfX09tba1TY1m+fDmp4eGMTUzs9P7RCQkAfPfdd84MS1hEZWXlSes0RdoKKUe3SNXW1vKrX/2KcaNHs7esjMWXX85zV1xBsJ+fQ4+bGRNDYmioFFLC7dhn7Bs5cqTZoXTLz8eHB887j6Xf/z47d+5k5syZ0kosHOLAgQMApFiku6ujBfv5MTE5WSYR64aRLVLTgeuAc5RSW2w/Fxi4f4/U1RgpcO6ivAcOHOCjjz7ie6NGdbl+Qkp4OFFRUVJIGUAp9bxSqkQptd3sWHqqY9e+qMBAwLGF1IoVKxg1ahS/+93vuGzUKDbccQdXjRnjlDU+lFKcmZHB6tWraWlpcfjxRP+4Yk6ZxT72L33rVpMj6bmLhg/n3YULqaysZObMmezbt8/skISb2bNnDyF+fsQ7sKeD1czIyGDjxo1UV1ebHYplGTlr39daa6W1HqO1Hmf7+dCo/XsirXWXLVKAUyeceOqpp/Dy8uLWyZO73EYpxfjx46WQMsYSYJ7ZQfRGx659UQ5skSosLOSKK67gggsuwNfXl08//ZTFl19OTHCw4cc6nbMGDqS0tFQmnXANS3CxnDLLF198gQImGbj2mjNMSE7m/e9/n7rycmbPnk1RUZHZIQk3smfPHgZHR7v9YrztzUxPp7m5ma89ZFxYX1hvFKlo09jYCNDpGClwXotUZWUlzz77LJcPH05yN03a48ePZ/v27W2xi77RWn8JmLfqch84o0WqsbGRJ598kuHDh/PBO+/wq3PO4asrr2TCl18adozemGUbJ/X2//2fKccXPeeKOWWWTz/9lNEJCW0XQ1zJiPh43rj2Wo4ePcqFF14o600Jw+zevZvB0dFmh+FUk1NT8ff3Z9WqVWaHYllSSFmYfQairgopZ7VI/fa3v+X48eP8aNq0brcdun8/DQ0N7Ny50wmRCassKdDY2Ehtbe1JLVIBvr4E+foaNtnEmjVryMrK4t5772V6YiJrf/xj7j3zTPw65IczJYWFMSU1lSWbNtHc3GxaHMIYVsknM9XW1vLNN99wpm2RdVc0ITmZJfPns23rVhZkZUnXW9FvdXV15OfnMzgmxuxQnCrQ15czU1P54IMPzA7FsqSQsrD6+noAU8dIPf/88zz55JPcMmkS45KSut3ePhHFpk2bHB2awDpLClRWVgKc1CIFrd37+tsidfToUW6++WZmzJhBeXk5b775Jq9efTXpkZH92q9RfjR1KgfKy3nvvffMDkX0k1XyyUzffPMN9fX1Ll1IAZw3ZAi/mzOH93ft4pFHHjE7HOHicnNz0Vp7XIsUwNzMTPbu3cuePXvMDsWSzLuUK7plb5EK6OMYqaqqKnbv3s3+/fspLCykuLiY8vLytkVSlVL4+voSGhpKREQEId99R+JVVxEcHMyRZctYtWcPK/fu5ZxBg/jDvJ4NLRgcHU2Inx+bNm3ixhtv7MOzFq6oqqp1ybj2LVLQ2r2vr4VUc3Mz//73v7n//vuprq7mrunT+emZZxKybRtYqI/6hcOGkRoezpNPPsmll15qdjhC9MuKhx7CWymmDRhgdij99sOpU9ly+DAP/vrXjNm7l/kvvWR2SMJF2YsITyyk5mRmwocf8v7773PPPfeYHY7lSCFlYfZCqmPXpVB/f7yV4vA778C99wKt3THWrVvHN998w/r169m8eTMHDx486XF+3t5EBgYS4ueHn7c3GmhobqamoYGKujrqmprgk0/ato8NDubnZ53FHdOm4dOhVawrXl5ejE1MZOPGjf145sLVGN0i9e2333LnnXeyadMmZqSn88eFCxkWF2dIrEbz8fZm0dSpPLByJW+//bYUU8Klfb5/PxOTk09aysBVKaX480UXkX34MLcsX87kxx4jsYvlO4Q4HXshNcgDC6m0iAhGjRolhVQXpJCyMHvXvo4tUkopksPDeW7DBqqnTCE3JIQ1a9a0bT8kOpoZc+YwZswYUrdsISMqiuSwMMIDArqcbUZrzYmGBirr6jjR0EBYQAAxQUE9LqDaG5+UxL83b6axsRFfX99eP16AUuoV4CwgRilVCDyotX7O3Ki6Zi+kOn75igoMZHsvCqmioiJ+8Ytf8NJLL5EYGsqz3/veaafct4pbJk3iv9u2ccPChXy1aBGj//xns0MSHbhaTpkhOzubTYcO8dvZs80OxTDBfn4sueoqznrmGa677jpWrVqFl5eMahC9s3v3bhITE93iAkNfXHTRRfzxj3+kvLycSIt0q7cK+d/EwtpapDopZpYtWMCUtDQWr19PcU4ON48fz6sLF5L3s5+x4c47+dfQofywvp6Lhg9nZHw8EYGBp/0yqpQixN+f5PBwMmNjSQgN7VMRBa2FVH19PTk5OX16vACt9UKtdaLW2ldrnWL1L3z2Qiq8Y9e+HrZI1dXV8eijjzI0I4NXX36Zn8yYwYY77uCK0aMtX0RB64QwL1x5JVprFixbxqFDh8wOSXTgajllhr///e8E+Phw7fjxZodiqKGxsTx2/vmsXr2aJ554wuxwhAvauXo1gzy0iAKYd+wYTU1NvPHGG2aHYjnSImVhXbVIAYxKSOC1q6+msbkZ3z4WPI4y3jYpxcaNGxk3bpy5wQinOF3XvmPHjlFbW0ugbWxfey0tLbz66qvcf//95Ofnc+GwYfxuzhwyoqKcEreRMqKi+M/3v8+1r77K1KlTWb58OZNPs+6aEFZSXl7O0qVLuXL0aJec9rw7102YwOp9+/jlL3/Jeeedx8SJE80OSbgIrTV7S0u5ZMQIs0MxzdjERIZER/Pyyy9z6623mh2OpUiLlIV1NUaqPasVUdD6hTI8PFzGSXmQtkKqQ4vUzPR0tNa89tprpzzm008/ZfLkyVxzzTVERUXx7g9+wMsLFrhkEWU3a+BAPrjxRpRSTJs6lXtmzqSmpsbssITo1lNPPUVNTc1pF113ZUop/nLRRcTHx3P11VfL+lKixw4cOMCx2lpGxcebHYpplFJcNWYMX3zxBQUFBWaHYylSSFnY6VqkrEwpxbioKNa9+67ZoQgn6apFakZ6OkNjYvjbAw+03bZp0ybmzp3LueeeS/Hevfzr0ktZffHFLj/dst2YxES+uvpqFowdy5+//pphw4bxyiuvyFo2wrI2btzII488wvdGjWKMG0/GEBkUxL/OO4+9e/bwk5/8xOxwhIv40rbg+xluMJNlf1w5ejQAy5YtMzkSa5FCysJON0bK6iYmJ7O9uLhtWmzh3iorKwnw8Tml9VQpxc2TJrG5qIg///nPXDhsGFlZWWzcuJGH58xhw513snDcOLcb/B0RGMg/L72UD2+8kYjGRq6++momT57Mxx9/jNba7PCEaHPs2DGuvfZa4uPj+dOFF5odjsOdmZHBXdOn8+9//5s333zT7HCEC/jqq6+ICAhgmIeuLWeXHhXFtLQ0nn32WVmAvh33+vbiZly1RQrgrEGDaGpp4bPPPjM7FOEElZWVp7RG2S0YO5ZgX1/uuecevsnP5xdnncWW227jzjPOINDNZ3U8Y8AAvrjtNp6+7DKKc3OZM2cO5557LmvWrDE7NCEoLi7mrLPO4kBuLk+fdx4RnYxjdEf3n3024xITueWWW6SbkujWl19+ybQBA9zugl9f3DZ5Mvv27eP99983OxTLkLPCwo4ePQqcOu7EFUxNTSXY15eVK1eaHYpwgsrKyi7P07CAAN687jreuu469v70p9x31lkueU73lZeXFwvGjmXTnXfyh3nzyFm/nhkzZjB37ly+/fZbs8MTHmrFTTeRlZnJvl27eO3qq5npJl1re8LPx4fnrriCRltrcVNTk9khCYs6cuQIe/fu9fhufXYXDx9OWloaf5YlPtpIIWVh69evJzE0lPiQELND6TU/Hx9mZGRIIeUhTtciBTAlLY2zBw2y5OQozuLv48OiqVPZctdd/Hb2bDZ9/TVnnHGGFFTCqaqqqrj77ru5aMkS/Hx8+PDGGzlr0CCzw3K6QdHRPPPMM6xZs4YH2o3hFKK9r776CoDpUkgBrQvQ33nnnXzxxRcyoZiNFFIO9PzzzzNy5EgaGhr69Ph169YxMTnZJdbR6cy5gwaxf/9+cnNzzQ5FOFhlZaXHLlTYW0F+fvy/6dPZaiuoNm/ezBlnnMGcOXOky59wmKamJhYvXsyQpCT++tRT3DhxIl/cfjvjbMtVeKIL9+7lpqwsHn/8cZYvX252OMKCPv74Y4J9fRmTkGB2KJZxVVkZkYGB/PznP5cxv0gh5VDvvfceO3bs4KOPPur1Y0tLS8nNzSUrJcUBkTnHuYMHA0irlAeoqqryqO56Rgjx928tqG67jYdnz+a7NWuYMWMGc+bM4ZtvvjE7POEmmpubeeWVVxg5ciS33347A6OiWH3rrfzpoovk4gfw6Lx5ZCUnc8PVV7N161azwxEWUlFRwbJly7h05Eh8PLg3RUfhAQHcN2sWq1evZsWKFWaHYzoppBxow4YNQN+mily/fj3QOvudqxoYFcWw2FieffZZuWrh5rrr2ie6Fuznx53Tp7P17rt5eM4cvluzhunTpzN37ty2/weE6K26ujqee+45RowYwdVXX42fnx9Lv/99Vtx0ExNc+HPFaP4+Prz0/e8T5u/PRRddRFFRkdkhCYt44YUXOHHiBLdNmWJ2KJZzU1YWg6Ki+MkPfkBtba3Z4ZhKCikHOXz4MIcOHSLM3593332X6urqXj1+3bp1eHl5Md6Fu10opfjxtGls2bKFTz75xOxwhIPU19dz9OhRYoKDzQ7FpQX7+XHnGWew9e67eei889i0aRNTpkzh4osvZvPmzWaHJwz0/vvvc2NWFt/86EeG7/vQoUM8+OCDpKenc8sttxBQVcWLV17Jl5dfzkXDh7tsV3FHSgoL49Wrr6a8vJx58+ZRVlZmdkjCZM3Nzfz9739nxowZjHXjtdX6ys/HhycuuIA9paXcd999ZodjKimkHGTTpk0A3DNzJrW1tbz99tu9evy6desYNWoUIS5+lf+qMWNITEzkscceMzsU4SAbN26koaHBpbuhWkmwnx93zZjBlttu41fnnMOaNWuYOHEil19+OdnZ2WaHJwxw//33s2TTJqb/61/8+9//7vf+6uvreeutt7jkkktIS0vj4YcfZmxYGO9cfz1f3H4780eOlKmbuzEmMZGXL7+cPTt2MHvMGI4dO2Z2SMJEf/zjH9m/fz+3ShHVpXMGD2bRlCn87W9/49133zU7HNPI/6wOsnHjRry8vLh50iQGDRrEP//5zx51b8vLy+N3v/sdX65ezXgXXD+qI38fHxaNHMnq1at54403zA5HOIB9VqNpaWkmR+JeQv39uffMM9ly++2t/dE//JAxY8Zw1VVXSUHlwrKzs8nOzub+s89mamoqv/nNb9rWDOyN+vp6VqxYwc0330xiYiKXX345GzZs4O4zzuC7O+/k9WuuYdbAgdIC1QtnDRrES9//PjnFxcycOZODBw+aHZIwwapVq7j//vu5bORILhkxwuxwLO03553H2MREFixY4LFje6WQcpCNGzcyfPhwQv39ueeee1i7di1ffPHFKdudOHGCjz76iHvuuYcRI0YwcOBAfvWrXzElLY27Z8wwIXLj3TZlClNSU7n22mvbvnQL9/HVV18xbNgw6drnIOEBAfzi7LPZetdd3DtzJh+98w5jxoxp++IsXMsrr7yCt7c3N06cyM9mzaKoqIiXX36528dprdm3bx+LFy/msssuIyYmhgsuuIA3Xn6Z2ampvHHNNWy/9VZ+fd55pEdFOeGZuKe5mZm8ce21FBYWMnny5E4/t4V70lrz17/+lUsuuYRhMTH8ff58uRDRjQBfX9645hqSAgO54Nxz+fzzz80OyemUWZMAZGVlaXedg15rTUJoKOcOHsy/LruM2sZGxv7lL4yZNo1nnnmGnTt3smbNGr744gs2bNhAY2Mj/t7enDFgALOHDOGCoUPd7oOw7MQJzn/nHfbt28cdd9zBvffeS2pqqtlh9ZlSapPWOsvsONozI6eam5uJiopiwYIFPO7C4/lcSXlNDf9cu5bF69dTWVfHrFmzWtcFuugifFy0FdtT8klrzYDISIbGxvLGtdeitWbWM89QHx7Oxo0bCWm3ZmBjYyPZ2dmsXbuWb775hq+++oqCggIAUsLDmT14MBcMG8aZGRn4u+j7bmU7iou5YfVqcnNzue+++/jVr35FYGCg2WH1mNVyysrf+aqrq1m+fDl/+tOf2L59O3MzM/n7JZcQ64JreJqloKKCK5YuZX9FBU888QR33nkn3m400+Hp8kkKKQfYsWMHI0eO5PHzz2+b7eUvX3/Nb9pNuOCtFOOTkpiRns6sgQOZkppKkJ+fWSE7RdmJEzz86ae8uGkTGpgyZQpnn302Z5xxBllZWSS6UF9kq31IgTk5tWXLFsaPH88zl13G98eOdeqxPV1VXR0vbd7M4l27KCgoICUlhRtuuIHrrruOzMxMs8PrFU/Jp5UrVzJv3jyevuwyFtjy5YNdu7jm1VcZMGAACxYs4MiRI2zfvp3t27e3dfmLCw5m2oABTB8wgFkDB5IZEyNXyp2gur6e+1asYNmWLWRkZPDggw9yzTXXuMQFC6vllJW+82mtyc3N5dNPP2XFihWsWrWK2tpahsfGcveMGVw1ZozkVx9U1tVx+5tv8tGePUyaNInHHnuMs846yy1eS6cUUkqpecBTgDfwrNb6D6fb3kpJZbSFCxfy3vLlbLnrrrYrGnWNjSz97jsCfX1Ji4hgQnIywW5eOHUl79gx3ty+nZV79rC5qIimlhYA4uLiGD16NCNHjmTo0KFkZmYyePBgUlNTLXdlwxkfUo7KKXvO9/c/t/Lycu6//36efvppsu++m9SIiH7tT/RNU3MzH+3ZwwsbN/LZ/v20aM2YMWOYP38+c+bMYdKkSfhbfNIaV86nnmppaWHChAmU5+ez4Y47TmpF+jY/n3s/+IDdR48SFxLC0NhYRsXHMz4piYkpKQyIiHCLLyOu6su8PH65ciXbjhwhOTmZm266iSuuuILRo0db9n2xWk6Z9Z2vqamJvLw8duzYwbZt29i0aRNr166luLgYaG3dPT8zk8tHjWJqWppl309XobXmjexsfrlqFcXHjzN+/Hiuu+46Lr30UjIyMswOr88cXkgppbyBPcBsoBDYACzUWu/o6jHuWkhlZ2czZswY7pkxg1+fd57Z4VjeiYYGso8c4buiIrYfOcLOkhJ2Hz3KicbGtm18fHxITU0lLS2N1NRUkpOTSUpKIiEhgfj4eGJiYoiOjiYqKgo/JxWnjv6QckROaa155ZVXuO+++ygsLCQ0NJQrrriC+fPnExcXR0tLCydOnODEiRNUV1dTVVVFZWUlFRUVlJeXU15eTllZGWVlZRQXF1N69CgaOH/oUF5ZuNDgV0D0RVFVFW/l5PD+zp2sKyykpaUFf39/Jk6cyPjx4xkxYgSDBw9mwIABJCUlERISYokvDq6YT721ZMkSbrzxRp793ve4YvToU+7XWqO1ltn1LEprzUd79vDchg2szs1FA0lJSUyfPr0ttzIyMkhJSSEyMtL0vLJaThmZTw0NDW2fUfbPp7KyMkpKSjhy5AiHDh3i4MGD7N+8mYKKiraLtQoYFB3NxORkpqSmMiM9nSHSuusQdY2NvLJ1Ky9s3Mi2I0cAGDBgAJMmTWL06NEMHjyYtLQ0kpKSiImJITQ01NLvgzMKqWnAb7TWc21//wJAa/1oV485XVItW7aMVatWnfaY3cXd2f32D6qOt7XYksx+f8eflpaWU36am5tP+mlqaqK2tpb8/HxaWlrYumgRkUFBp41RdE5rzeHqanLLyjhw7Bj5FRXkl5dzqKqKwspKjlRX02h7zzoKDAwkLCyMkJAQQkJCCAwMJCAgAD8/P/z8/PDx8cHHxwcvL6+2H6VU2w+0ttRkZmbywAMPdBmjEz6kDMup/fv3c9999/Hll19SUlLC+KQk5gwZQmFlJW/n5JxUtHYmwMeHiIAAIgMDiQgMJDooiNjgYBJCQ5k3dKissWFRFbW1fJmXx9qCAr4rKiL7yBGONzSctE1AQACRkZGEh4cTHBxMUFAQ/v7++Pn54evri7e3N97e3qfNlc4+/Dq77Xe/+x0pXUyR70r5BPD73/+enTt3tn2etP9saf9vS0sLTU1N5Ofns3nzZiYkJfHJLbdIseTijlRXs2rPHr7Iy2P9wYMcrKw86X4fHx8iIiK6zCsfH5+Tcqt9XnWWY3btfz/33HO5/vrru4zRajl1unz67LPP+P3vf09eXh5lZWXU19cTHBxMSEgI3t7eNDU1UV9fT01NDTU1NTQ1NXUZl5dSxAUHkxIeTmpEBBmRkQyKjmZobCzDYmNdfkkZV5RbWsqn+/axJj+f7Lo69u/ff8o2Xl5ehIWFteVLQEAA/v7++Pr6tuWM/btbZ59JXeURcNrPK/vf3t7eLFmypMvn4IxC6gpgntb6Ftvf1wFTtNZ3dNjuNuA2259Dgd39PrhjxAClZgdhAHd4HlZ9DgO01rGO2rkb5lRPWfX9diR5zpJPveWJ5wx47vOG3j9303PKyfnkKeeGPE9zdJlPRo2Y7Kw97pQKTWu9GFhs0DEdRim10UqDNPvKHZ6HOzyHPnKrnOopT3y/5Tk755Cd3Oay+eSJ5wx47vMGSz73bnPKmflkwdfHIeR5Wo9R/QsKgfZzWacARQbtWwhPJDklhHEkn4QwluSUEBhXSG0AhiilMpRSfsAC4F2D9i2EJ5KcEsI4kk9CGEtySggM6tqntW5SSt0BrKR1GszntdY5RuzbJJbv2tFD7vA83OE59Job5lRPeeL7Lc/ZwdwwnzzxnAHPfd5gseduwZyy1OvjQPI8Lca0BXmFEEIIIYQQwlXJHKxCCCGEEEII0UtSSAkhhBBCCCFEL0khBSilopRSHyul9tr+jexiuwNKqWyl1BallHFL3veDUmqeUmq3UipXKfXzTu5XSqm/2u7fppSaYEacp9OD53CWUqrS9rpvUUr92ow4hXHc4bztLU88z5VSzyulSpRS27u43+3eZ6N4Yo7YeWKugORLT3hKXnhCDrjN+W5fkd2Tf4DHgZ/bfv858FgX2x0AYsyOt1083sA+YCDgB2wFRnTY5gJgBa1rPkwF1pkddx+ew1nA+2bHKj9Ofc8tfd466Dm73XkOnAlMALZ3cb9bvc9OPl/c8rXz1FyxPS/Jl/6fGy7/GnlKDrjL+S4tUq3mAy/afn8RuNS8UHplMpCrtd6vtW4AXqX1ubQ3H3hJt1oLRCilEp0d6Gn05DkI9+IO521veeR5rrX+Ejh2mk3c7X02iifmiJ1H5gpIvvSAp+SFR+SAu5zvUki1itdaHwaw/RvXxXYaWKWU2qSUus1p0XUtGTjY7u9C22293cZMPY1vmlJqq1JqhVJqpHNCEw7iDudtb8l53jl3e5+N4ok5Yie50jV3fc97ylPyQnKglUu8l4asI+UKlFKfAAmd3PVAL3YzXWtdpJSKAz5WSu2yVdRmUZ3c1nE++55sY6aexLcZGKC1Pq6UugB4Gxji6MCEw7jDedtbcp53zt3eZ6N4Yo7YSa50zV3f857ylLyQHGjlEu+lx7RIaa3P01qP6uTnHaDY3lxo+7eki30U2f4tAd6itfnVTIVAaru/U4CiPmxjpm7j01pXaa2P237/EPBVSsU4L0RhMHc4b3tLzvPOudv7bBRPzBE7yZWuuet73lOekheSA61c4r30mEKqG+8CP7D9/gPgnY4bKKWClVKh9t+BOUCnM4040QZgiFIqQynlByyg9bm09y5wvW32k6lApb0bo0V0+xyUUglKKWX7fTKt522Z0yMVRnGH87a35DzvnLu9z0bxxByxk1zpmru+5z3lKXkhOdDKJd5Lj+na140/AK8rpW4GCoArAZRSScCzWusLgHjgLdt56wMs01p/ZFK8AGitm5RSdwAraZ3l5XmtdY5SapHt/qeBD2md+SQXqAFuNCvezvTwOVwB/FAp1QTUAgu01pZr3hU94w7nbW956nmulHqF1tmlYpRShcCDgC+45/tsFE/METtPzRWQfOmOp+SFp+SAu5zvysVedyGEEEIIIYQwnXTtE0IIIYQQQohekkJKCCGEEEIIIXpJCikhhBBCCCGE6CUppIQQQgghhBCil6SQEkIIIYQQQohekkJKCCGEEEIIIXpJCikhhBBCCCGE6CUppIQQQgghhBCil6SQEkIIIYQQQohekkJKCCGEEEIIIXpJCikhhBBCCCGE6CUppIQQQgghhBCil3zMOnBMTIxOT0836/BC9MumTZtKtdaxZsfRnuSUcFWST0IYq7ucUko9D1wElGitR3VyvwKeAi4AaoAbtNabbffNs93nDTyrtf5Dd/FIPglXdrp8Mq2QSk9PZ+PGjWYdXoh+UUrlmx1DR5JTwlVJPglhrB7k1BLg78BLXdx/PjDE9jMF+BcwRSnlDfwDmA0UAhuUUu9qrXec7mCST8KVnS6fpGufEEIIIYQH0Vp/CRw7zSbzgZd0q7VAhFIqEZgM5Gqt92utG4BXbdsK4ZGkkBJCCCGEEO0lAwfb/V1ou62r20+hlLpNKbVRKbXx6NGjDgtUCDNJISWEEEIIIdpTndymT3P7qTdqvVhrnaW1zoqNtdQQSCEMY9oYKdG1yspKZs2axdChQ3nssccwY4BmY2MjhYWF1NXVOf3YVhIQEEBKSgq+vr5mh+LWDh48SFVVFSNGjKB1jLN7kXxqJfkkjCI51cqBOVUIpLb7OwUoAvy6uF24MMmnVn3JJymkLOhnP/sZ2dnZ7N27lw/ffpud+/aRkpLi1BgKCwsJDQ0lPT3dLb/Y9oTWmrKyMgoLC8nIyDA7HLfV1NTErFmzyMvLIyMjgxUrVjB06FCzwzKU5JPkU39UPvQQAOEPPmhyJNYhOeXwnHoXuEMp9Sqtk01Uaq0PK6WOAkOUUhnAIWABcLXRB+9O5UMPST4YSPKp7/kkXfss5osvvmDx4sXcc889bNiwgeMNDbz0UleT6jhOXV0d0dHRHptQAEopoqOjPf4KjaO99dZb5OXlsWjKFI4cOcJTTz1ldkiGk3ySfBLGkpzqX04ppV4BvgWGKqUKlVI3K6UWKaUW2Tb5ENgP5AL/Bn4EoLVuAu4AVgI7gde11jn9fzbCTJJPfc8nKaQs5o9//CPJycnc4+dH8n//y7S0NF566SW07rQLskN5ckLZyWvgeE/ccw8Do6J4ZO5cLho8mFdeeIH6+nqzwzKcnEvyGvRFfX09/922jZqGBrNDsRw5n/r+GmitF2qtE7XWvlrrFK31c1rrp7XWT9vu11rrH2utB2mtR2utN7Z77Ida60zbfY8Y9FR6rKGhgYraWrf8nDCT5FPfXgMppCyktraW1atXc/nllxPk5wfAwnHj2L17N+vXrzc5OiGMt3btWjYUFrJoyhS8vbxYOHYsFXV1vPfee2aHJoQl/Otf/+LWN9/k3H//m507d5odjhCmqq+vJzU1lfTHHiMhIYHy8nKzQxIeTgopC/nss8+ora3lrHb/McwfMYKAgABefPFFEyMTwjHee+89fLy8WDhuHACzBg4kKTRUznchbJYsWUJ6ZCRHT5zgyiuvNDscIUx14MABSkpKmD14MBUVFaxYscLskISHk0LKQt76zW8I8vVl+oABbbeFBwQwd+5cVq1aZWJkQjjG2rVrGRUfT6i/PwDeXl5cNmoUK1eulLE0wuNt2bKFrVu3cse0adx75pnk5OSQl5dndlhCmMZ+/t89YwZxcXHSe0GYTgopi9Bas3LPHmYNHEhAh2kXJ5w4wb59+ygtLTUpOnPde++9jBgxgjvvvLPP+6itrWXWrFk0NzcDrTPUvPbaa0Brf+szzzyTpqYmQ+IVPdPc3MyGDRuY1GFGyimpqTQ2NrJlyxZzAnNzkk+uY8mSJfj5+fG9UaM4b/BgAD766COToxIdSU45j72QSo+M5MILL2TFihU0NjaaHJUwkqvlk2GFlFIqQCm1Xim1VSmVo5R6yKh9e4Jdu3ZxsLKSuUOGnHJflu2L5rp165wdlun279/PmjVr2LFjB3/729/6vJ/nn3+eyy+/HG9vbwBWr17N5s2bAfDz8+Pcc89tSzLhHDt37qS6upqJHQopTz7fHU3yybW89tprzBs0iMigIAZHR5MWESGFlMVITjlXXl4efn5+JIaGcvaJE1RWVvL111+bHZYwiCvmk5EtUvXAOVrrscA4YJ5SaqqB+3dr3377LQDTO1l8d1xSEt5KsXbtWidHZa7du3cza9Ys8vPzGT9+PCdOnOjzvl5++WXmz58PwNdff80999zDG2+8wbhx48jLy+PSSy/l5ZdfNip00QP287lji1RSWBjJyclSSBlM8sm1HD16lCNHjjAlLQ1onU1q9uDBrF69mgaZwc8SJKec78CBAwwYMAAvLy/OGTQIP29v6d7nJlw1nwxbkFe3zs993Panr+3H+XN2u6gNGzYQ5u/PoKioU+4L9vNjZHy8aYXU3XffbXg3q3HjxvGXv/zltNsMHTqUH/zgB6Snp3PLLbf0+VgNDQ3s37+fdFuROmPGDCZNmsQf//hHRo0aBfyvm5lwnrVr1xIVFcXATs75CWFhbltIST6JnsjJaV2aZ3hsbNtt5w4ezHMbN7JmzRrOPvtss0KzHMkpz2FfuB0gxN+fswYOZPny5fzxj3/Ey0tGqxhB8ql3DD3rlFLeSqktQAnwsdZ6XYf7b1NKbVRKbTx69KiRh3Z5GzZsYFxSUpf/EUxKSWHdunVt/T09RXZ2NmPHjgXgrrvuoqamhv3793PzzTdzxRVXAK1dxBYtWsQVV1zBv/71LwA2bdrEM888A0BpaSkREREn7Xf37t0MHTq07W9vb2/8/Pyorq52wrPqnid0lV27di1Tp07tdN2GrORk9u/fj/w/Yaye5BPAiRMnmDhxIu+//z7g+vnkiuyF1LC4uLbbZmZkoJTiyy+/NCss0UFPcurzzz9n5syZLFq0iM8//xyQnOqr/Tk5JJWVtf19+ahRFBQUeFyPHXfVk3xqaWnhgQce4M4772yb4dfMfDKsRQpAa90MjFNKRQBvKaVGaa23t7t/MbAYICsrS1qrbOrr69m2bRs/njy5y22yUlJ4buNGdu3axciRI50YHd1eRXCknJwcRo4cybFjx1BKERQUxMCBA3nuuefakmr48OE8/fTTtLS0cOuttwIwceJE/vznP3P77bcTGBh40gxwZWVlhIeH49thUo/6+noCAgKc9+ROz95V9rhSyhf4Wim1QmvtFp8WJ06cYEdODhe3u9renn3c1Pr167nwwgudGZrDWT2fAB577DGuuuqqtr/dIJ9czo4dOwgLCyMxNLTttlB/f4YMGcLWrVtNjMx6rJ5TSilCQkKoq6sjxfZ/m+RU71VXV3OstpYB7b4kXzB0KAEBAbz66qucccYZ5gXnRqyeT++88w6HDh0iKirKEvnkkHZQrXUF8DkwzxH7dzdbt26lsbGR8cnJXW4z0XafJzXtV1dX4+vrS1BQEFu2bGHEiBFdbvvuu+8yY8YMzj333LbbAgICKC4uJjIykubm5rbEysvLIykp6aTHl5WVERsbe0qimcW2qrzbdpXds2cPGhje7mp7e+NtrbOyELVxeppPn3zyCSNGjCA+Pv6k2105n1yR/QtFxxbbET4+bP7sM5OiEu31NKdmzpzJihUreOyxx3jwwQfbbpec6p0DBw4AMCAysu22sIAA5mRk8Prrr5/SY6exsZE///nPTE9PJyEhgYsvvpi3337biRGL3uhpPu3evZtp06bx5JNPtvVCAvPyychZ+2JtLVEopQKB84BdRu3fndmLowkd3uj2BkVH4+/v39bdwxNs3769rT/rsWPHTmmqbe+SSy7hm2++OWnwYGRkJFVVVQDMmTOnbWafYcOGUVpayqhRo/jmm2+A1sWQL7jgAgc9k77prqusbRuX7C67a1frfw2ZMTGd3h/s58eQIUPIzs52Zlhuraf59Nlnn7F27VqWLVvGv//9b1paWgDXzydXYy+kOhqdkEB+RQWVlZUmRCXa62lO2bvsR0ZGUl9f33a75FTv2Kc+H9Dhdf7eqFEUFxezbNmytts2b97MhAkTuOeee6hpaGDOnDls3bqVyy67jJ///Oe0DusXVtLTfEpJSSHSVkzbZ+UD8/LJyK59icCLSilvWgu017XW7xu4f7e1YcMG4uLiSAkP73Ibby8vhkREsPXtt+GJJ5wXnImmTZvGf//7XwAyMzPbpv0tKyvjgQce4LvvvuPRRx9l2rRpvPnmm9TX15+UGIcOHSLNNuPVHXfcwZNPPsl5551HSEjIKS0dy5Yt49FHH3XSM+uZ7rrK2rZxye6yu3btwkupTieasBs5ciTbt2/v8n7ROz3Np0ceeQRoXcMoJiam7Uugq+eTKzl69CilpaVkHDwIHWa1HJ2QAMC2bduYOXOmGeEJm57m1NChQ1m5ciUVFRXccccdbY+XnOodeyGV1uEL9vlDhzJ9+nRuvfVWfHx82LNnD4/89rfEBgfz8oIFXDhsGOEPPkhzczO3T53KY489RmNjI3/6059MeBaiKz3Np7vuuos777yTr776ijPPPLPt8Wblk5Gz9m0Dxhu1P0+yYcUKxoWHdzrovr3hcXGsLShwUlTWMnr0aP75z38CEB0dzdNPP33S/WedddZJfx8/fpywsDD8/f0BGD9+PGeffTbNzc0nXcGA1hleLr300pMGIlqJ1rpCKfU5rV1l3aKy2LVrF+mRkfj7dP1f0MDSUt7eu5f6+vq291EYo7t8ArjhhhvafnenfHIFnc3YZzfKVkht2bJFCikL6S6nLr/88pP+lpzqvQMHDhDk60tMcPBJt/v5+PDSGWcwe+dOrr76aqB17NTf588nKigIgMqHWudr+uOFF6KU4sknn2TOnDnMnTvXuU9C9Eh3+fTcc8+d9LeZ+SRzRZqsoaGBPaWljOowHqEzQ2NjOVhZyfHjx7vd1t0opbjmmmuoqanp0fZFRUX89Kc/Pem2m2666ZSEgtbF2a6//npD4jSKu3eV3bVrF0O66NZnNzw2lhat2b17t5Oi8hyelk+uprMZ++wSQ0OJDgqSCScsRnLK8QoLC0kOC+v0onN0cDCrbr6Z166+mu0/+QnLFi5sK6LaU0rx8Jw5jBw5khtuuIFjx445I3TRS66UT1JImWzXrl00tbQwogeF1DDb1ckdO3Y4OixLmjlzJkGd/MfYmczMTAYNGuTgiBwqEfhMKbUN2EDrGCm36Crb3NzMnj17uhwfZWf/EulJ4wKdycPyyaXs3buXED+/k2bss1NKMTohwfB1XkT/SU45VkVFBZGBgV3eHxMczNzMzNMOkwAI9PVl6dKlFBcX84c//MHoMIVBXCWfpJAymX0MyIguZi9rzz7DmacWUp5Ea71Naz1eaz1Gaz1Ka/1bs2MySkFBAXV1dd22SA2OjsZbKTnfhccpKCgg9TTdvUfFx7N9+3aampqcHJkQ5qmoqCDcoOnfx40bx/fHjOFvf/4zhw4dMmSfwjNJIWWy7OxsfL28GBwd3e226ZGR+Ht7yxV64dK6m7HPzt/Hh4FRUXK+C49TUFBw2qvqI+Ljqa+vZ9++fU6MSghzGVlIVT70EL846yyaW1p4+OGHDdmn8ExSSJksOzubITEx+J1m0L2dt5cXQ2Ji5IulcGk9LaSgdVygnO/C0xQUFJB8mkJqqC13NrZbk0gId2dkIQWt61FdP2ECL7zwAsXFxYbtV3gWKaRMlp2d3aNufXbD4+Kc1tVJ1lmQ18ARdu3aRVRgYKcDgTsaFhdHbm7uSWuvuCo5l+Q16Ina2lqOHj162hapTNt42d0utHacI8j55DmvgdaaiooKwgwspAAWTZ1KQ0MDzzzzjKH7dUWeci6dTl9eAymkTFRZWUlBQUGPJpqwy4yJIT8/v8czmfRVQEAAZWVlHp1YWmvKysoIMPg/bk+Xm5vLoB50ZYXWCVZaWlrYs2ePg6NyLMknyaeeOnjwIMBpC6lQf3+Sw8LYXVrqrLAsR3Kq/zmllJqnlNqtlMpVSv28k/t/qpTaYvvZrpRqVkpF2e47oJTKtt23sZ9PpVs1NTU0NTUZ2iIFMCQmhtmDB/Ovf/2LhoYGQ/ftSiSf+p5PRi7IK3qpNxNN2NnHUuXm5jJmzBiHxAWtK0cXFhZy1MOveAYEBJDSYUFM0T/79+9nsm1V8u7Yz/fvfvc7Rr/2miPDcijJp1Zm5ZNSKhV4CUgAWoDFWuunnB5IDxTY1grsbuaxzJgY9njw+SQ51aqvOaWU8gb+AcwGCoENSql3tdZtXV601k8AT9i2vxj4ida6/XzhZ2utnVLNV1RUABheSAHcPmUKV7z8Mv/973+55pprDN+/K5B8atWXfJJCykT2Qmp4LwqpgbYvlnv37nVoIeXr60tGRobD9i88U2NjIwUFBVw5Y0aPtm+7cODiV94ln0zXBNyrtd6slAoFNimlPm7/pdEqetIiBa3jB1/avJmWlha8vDyvc4nkVL9NBnK11vsBlFKvAvOBrnJiIfCKk2I7RWVlJeCYQuqcQYMYOHAgzz33nMcWUpJPfed5//taSE5ODiF+fqRFRPT4MYOiogBcvquT8EwFBQW0tLSQ3sMWqRB/fxJDQ8ktK3NwZMKdaa0Pa603236vBnYCyeZG1bk9L72EApI6WUOqvaGxsdQ0NrYVXkL0UjLQ/uQppIucUEoFAfOA5e1u1sAqpdQmpdRtXTzuNqXURqXUxv62dDiyRcrLy4sFAwbw2WefkZeXZ/j+hXuTQspEO3bsIDMmpsu1QjoT4u9PQkgIe/fudWBkQjjG/v37AXpcSAEMio5mnxRSwiBKqXRgPLCuk/sM++LXV4WVlcSHhHQ7k+tQD1+gXfRbZ188uhogczGwpkO3vula6wnA+cCPlVJnnrIzrRdrrbO01lmxtvO1rxxZSAEsHDsWBbz44osO2b9wX1JImWjHjh0M60W3PruB0dHSIiVcUl8KqcHR0dIiJQyhlAqh9ar63Vrrqo73G/nFr68KKyu77dYH/5sCfefOnY4OSbinQiC13d8pQFEX2y6gQ7c+rXWR7d8S4C1auwo6jKMLqdSICGYNHMiSJUtoaWlxyDGEe5JCyiTl5eUcPnyYYX34sB4cHS0tUsIl7du3Dz8/PxK76bbU3qDoaI7V1lImxZToB6WUL61F1Mta6zfNjqcrPS2kooODiQkKkhYp0VcbgCFKqQyllB+txdK7HTdSSoUDs4B32t0WbBtriFIqGJgDbHdksPZCKszf32HHuHrcOPLz81l1yy0OO4ZwP1JImcR+FXFoHwqpQVFRlJSUtA2+FMJV7H7/fQaEhvZqcPyQdhOsCNEXqrX/9HPATq31k2bH0xWtdY8LKWhdT0papERfaK2bgDuAlbSOGXxda52jlFqklFrUbtPLgFVa6xPtbosHvlZKbQXWAx9orT9yZLyObpECuGDoUAJ9fFi+3aE1oXAzUkiZJCcnB+hjISVfLIWLOlBe3qtuffC/mft2797tiJCEZ5gOXAec025dnAvMDqqjsrIy6pqaSO5hITXMVkh58tovou+01h9qrTO11oO01o/Ybntaa/10u22WaK0XdHjcfq31WNvPSPtjHan4gw/w9/YmwNfXYccI8fdn3tChvJ2TQ1NTk8OOI9yLFFIm2bFjB0FBQaT18AOzPXshJeOkhCvRWvepkBoQGYm3UnK+iz7TWn+ttVZa6zFa63G2nw/NjqujwsJCAJLCwnq0/dDYWMrLyykuLnZkWEKYrqq+3qGtUXaXjxpFaU0Nn332mcOPJdyDFFIm2bFjB8OHD+/T+h8ZkZEopaRFSriU8vJyqurrGdDLQsrX25v0yEgppITbKypqHevf3dTndvYJJ2SclHB3lXV1TimkZg8eTJi/P6+8YtqSWcLFSCFlkh07djBixIg+PTbA15eUsDByli/vfmMhLKIvM/bZDY6Olq59wu3ZC6mEnhZStq7hMk5KuDtnFVIBvr7MGzqUt155hcbGRocfT7g+KaRMUFVVRWFhIQOLuppptHsDo6LIO3as+w2FsAj7Qoe9bZGC1u6subm5Mi2tcGv2Qio+JKRH2yeEhhLm7y8tUsLtOauQArhk+HAq6ur4/PPPnXI84dqkkDKBfaKJvqwhZTcwKor9UkgJF9JWSEVE9Pqxg6Ojqa2tbfuiKYQ7KioqIiYoqNvFeO2UUgyLjZVCSri9yro6wpxUSJ07eDDBvr4sl14/ogekkDKBvZAa3o8FHwfa1tYpLy83KiwhHOrAgQNEBgb26cNQJlgRnqCoqKjH3frsZAp04Qmc2SIV6OvLnMxM3nrrLZqbm51yTOG6pJAyQU5ODkG+vqT14cq83cCoKAByc3MNikoIx8rLy+vzOS+FlPAERUVFvVqsGlrHSRUXF3NMeigIN6W1dmohBa3d+0pKSvj666+ddkzhmqSQMkFOTg6ZMTF9mrHPTgop4Wry8vL61K0PWmcxC/TxkUJKuLW+tEgNs/VskO59wl3V1dXR0Nzs1ELqvCFD8Pf35+2333baMYVrMqyQUkqlKqU+U0rtVErlKKXuMmrf7iYnJ6df46PgfzOfSSElXIHWmvz8/D63SHl5eTEwOlqm/Bduq6mpieLi4l4XUsNtnyXZ2dmOCEsI01VWVgI4tZAK9ffnvPPO46233pIFr8VpGdki1QTcq7UeDkwFfqyU6tv83m6soqKCoqKifo2PgtY+vMlhYVJICZdw5MgR6urq+jRjn93g6GhpkRJuq6SkhJaWll537UsOCyMiIoJt27Y5KDIhzFVRUQE4t5ACmOvtTX5+Plu2bHHqcYVrMayQ0lof1lpvtv1eDewEko3av7swYsY+u4yoKCmk3JS7tfAeOHAAoF/jAgdFRbF//35Z20O4pd6uIWWnlGLMmDFSSAm3ZVYhdcHQoXgpxVtvveXU4wrX4pAxUkqpdGA8sK7D7bcppTYqpTYePXrUEYe2PHs/9mH9bJGC1nFSUki5Lbdq4e3P1Od2g6KjaWpqaivKhHAn9kKqty1SAGPGjCE7O1vWWRNuyV5Ihfr7O/W4McHBTE1Lk0JKnJbhhZRSKgRYDtytta5qf5/WerHWOktrnRVrQCHhinJycggKCiI1PLzf+xoYFUVJSQlVVVXdbyxciru18NqLn9R+FFKDbTP3yTgp4Y762iIFMCQ/n+rqavLz840OSwjT2b/jhDm5kAK4cNgwtm/fzr59+5x+bOEaDC2klFK+tBZRL2ut3zRy3+4iOzubYRER/Zqxz05m7vMMXbXwupK8vDxig4MJ9vPr8z7shdTu3buNCksIyygqKsJLKWKDg3v92JHx8QDSvU+4JXsh5ewWKWgtpADeeecdpx9buAYjZ+1TwHPATq31k0bt151ordm6dSujEhIM2d8guULv9k7Xwmu73yW6yx44cKBf46MAooODiYqKkkJKuKWioiLigoPx8fbu9WOHxcaigK1btxofmBAmq66uBswppNIjIxkVHy/ToIsuGdkiNR24DjhHKbXF9nOBgft3eYcPH6asrKzt6mF/ZdhmQJOZzNxTT1p4XaW7bH/WkGpvSHAw2z/6qP8BCWExfVlDyi7E35+MqChpkRJuycwWKWhtlVqzZg0lJSWmHF9Ym5Gz9n2ttVZa6zFa63G2nw+N2r87sH/IGVVIBfn5kRIWJi1SbsidWnibm5vJz89vW/usP4bExLC3tNSAqISwlkOHDvW5kILWzxVpkRLuqKqqimBfX7wNGBLRFxcOG0ZLSwvvvfeeKccX1mbOWemh7IXUKIMKKYCBsraOu3KbFt7CwkKampqMKaSioyk5caJtFich3EVRURFJYWF9fvy4xERyc3MpLy83MCohzFdVVWVaaxTA6IQEBgwYIN37RKekkHKirVu3kpqaSkRgoGH7HBwdLS1SbsidWnj3798P0K/FeO2GxMQAMuGEcC/19fWUlpb2q0UqKyUFgA0bNhgVlnBzSql5SqndSqlcpdTPO7n/LKVUZbuLeb/u6WONZHYhpZTi0ksv5eOPP24bryWEnRRSTrTlk08YYWARBa0z9x07doyysjJD9yuEUYwspDJthdSuXbv6vS8hrOLw4cNA36Y+txuXlIRSinXrXHZyT+FESilv4B/A+cAIYGEXaxV+1e5i3m97+VhDmF1IAcyurKS+vp6VK1eaGoewHimknKS+vp49paWGdusDWVtHWN/+/fvx9vYmpR/dluwGREbi4+UlLVLCrdjXkErqRyEVHhDAsGHDWL9+vVFhCfc2GcjVWu/XWjcArwLznfDYXrNCITU1NZXooCBZnFecQgopJ9m5cydNLS2GTTRhZ58CXcZJCavav38/AwYM6NO0zh35enszMCpKWqSEW+nPYrztjff3Z926dWitjQhLuLdk4GC7vwvpfNH3aUqprUqpFUqpkb15rFHLc1ihkPLx9mZeZibvv/8+DQ0NpsYirEUKKSfZvHkz0Dpo0UgDIiLw9vaWFilhWXl5eQwcONCw/Q2JiZFCSriVQ4cOAZDYz0JqYnIyR48eJT8/34iwhHtTndzWsQLfDAzQWo8F/ga83YvHGrY8hxUKKYCLhg+nqqqKTz/91OxQhIVIIeUkGzduJMzfn0FRUYbu18/Hh/T0dGmREpa1f/9+Ywup6Ghyc3NpbGw0bJ9CmKmoqAg/Pz+igoL6tZ+Jya2NAjJOSvRAIZDa7u8UoKj9BlrrKq31cdvvHwK+SqmYnjzWSFYppM4eOJAQPz9eeeABs0MRFiKFlJNs2LCBsYmJeDlgHYTMzEwppIQlVVdXc/ToUUMLqWFxcTQ2NpKbm2vYPoUwU1FREUm2ySL6Y2R8PEFBQXz99dcGRSbc2AZgiFIqQynlBywA3m2/gVIqwbamIUqpybR+ZyzryWONorWmqqqKMAsUUgG+vszNzOSDXbtoamoyOxxhEVJIOUF9fT1bt25lQnJn3Y/7L6O8nN3bt9PS0uKQ/QvRV3l5eQAkbNpk2D5HxMUBkJ2dbdg+hTDToUOHSEpK6vd+fL29mTlzJqtXrzYgKuHOtNZNwB3ASmAn8LrWOkcptUgptci22RXAdqXUVuCvwALdqtPHOiLOmpoaWlpaLNEiBXDJ8OGU1dTw1VdfmR2KsAgppJxg+/btNDY2Mt6AD8rODI2NpbapSfrFC8tpm/o8IsKwfWbGxOClFNu3bzdsn0KYqXDbNmINWmR6RksLO3fubJvAQoiuaK0/1Fpnaq0Haa0fsd32tNb6advvf9daj9Raj9VaT9Vaf3O6xzpCVVUVgGUKqfOGDCHQx4c33njD7FCERUgh5QT2BRLHOaiQsq+ts3PnTofsX4i+MnINKbsAX18GRUVJISXcxpHq6n5PNGE3y9aNVlqlhDuwL4BrlUIq2M+P84YMYflLL9Hc3Gx2OMICpJBygm8WLyYqMNDQq/LtDbXNhiOFlLCavXv3EhEQ0O9B9B0Nj4uTQkq4herqaqobGgwrpEbFxxMdFMQnn3xiyP6EMJPVWqQALh85kuLjx6V7nwCkkHKKzUVFjDdgIHFXooKCiA0OlkJKWM7evXvbFo020vC4OHJzc6mpqTF830I4k1FrSNl5eXkxKyODTz75RNaTEi7PioXUnMxMgnx9ee2118wORViAFFIOVllZyc6SErJSUhx6nMyYGHbs2OHQYwjRW3v27GlbNNpII+Lj0VrLxQPh8uyFVGJYmGH7PGvQIIqKiti6dath+xTCDFYspIL9/Jibmcny5ctl9j4hhZSjffPNN7RozRkDBjj0OENjY9m5c6dcgRSWUVNTw8GDBx1TSNlm7pPufcLV2RfjTQgJMWyfFwwdirdS/OfOOw3bpxBmsGIhBXDZyJEcPXqUzz77zOxQhMmkkHKwr776Ch8vL6e0SFVUVFBcXOzQ4wjRU/Z1nhzRtS8jMhJ/f3+ZAl30mFLqeaVUiVLKUtX3wYMHAUgODzdsnzHBwZyZkcFbOTlycU24NKsWUrOHDCE0NJRly5aZHYowmRRSDvbll18yPimJYD8/hx5HJpwQVrN3714Ah7RI+Xh7M2rUKLZs2WL4voXbWgLMMzuIjgoKCogKDDT8M+KyUaPIKy9n8+bNhu5XCGeyaiEV6OvLJYMGsXzZMmpra80OR5hICikHqq2tZcOGDUxLS3P4seyF1MYnnnD4sYToiT179gAwMCrKIfsfDWxcs0auuIse0Vp/CRwzO46OCgoKSDGwNcru4mHD8PHykgHxwqVVVVXh5+2Nv4+P2aGc4qoxY6huaOC9994zOxRhIimkHGj9+vU0NDQ4fHwUQGJoKBEBAeRI1z5hEXv27CExMdFhVxLHJyVRWVfXtlaVEP2llLpNKbVRKbXx6NGjTjlmfn4+qQ5YGiMyKIjZQ4bw4osvUl9fb/j+hXCGqqoqy7VG2c1ITycpNJSlS5eaHYowkRRSDvTFF1+ggKlOaJFSSjE6IYHtUkgJi9i5ejUDHdildWxiIgCbNm1y2DGEZ9FaL9ZaZ2mts2JtrfwOPh75+fkOaZECuH3yZEpKSnj11Vcdsn8hHM3KhZS3lxdXjB7NihUrKCkpMTscYRIppBzoo48+YnxSEhGBgU453qiEBHYUF8tq28IScsvKHDI+ym5EXBx+3t5s3LjRYccQwpEqKys5fvy4wwqpWQMHMiIujr/85S/SBVa4JCsXUgALx42jqalJWqU8mBRSDlJWVsa6deuYPWSI0445OiGB2qamtkH+QpilvLycspoaBjlofBSAn48PI+PjpUVKuKyCggIAUh1USCmlWDR1Klu2bGH16tUOOYYQjmT1Qmp4XBxZyck899xzcrHCQ0kh5SCrVq2ipaWFOU4spEbFxwPITGbCdDk5OQAMs6335CjjEhPZtGmTfICJbimlXgG+BYYqpQqVUjebHZOjCymAq0aPJjU1lfvuu4+WlhaHHUcIR7B6IQVw7fjx7Nixgw0bNpgdijCBYYWUVdfoMMuHH35ITEwM45OSnHbMYbGx+Hp5yWr2wnT29Z1GOLiQGp+URGVlZduaVUJ0RWu9UGudqLX21VqnaK2fMzsmeyHlqK59AAG+vvwyK4vNmzfz8ssvO+w4QjhCdXU1YRYvpC4bNYrAwECeffZZs0MRJjCyRWoJFlyjwwwtLS189NFHzJs3Dy8v5zX6+fn4MDQ2VgopN+DqFyays7MJ8/d36BdEgAnJyQB8++23Dj2OEI6Qn5+Pn7c3scHBDj3OlaNHMz4piV/84hdUVlY69FhCGKm8vNzyhVR4QADfGzaMl5cskfzyQIZ9y7fqGh1mWLNmDaWlpZxtwiJtoxISpGufe1iCC1+Y2L59O8Pj4lBKOfQ4I+LiCA8IYPVTTzn0OEI4QkFBAclhYQ6/4Obl5cUTF1zA4aIi7r77boceSwijNDc3c+zYMaKCgswOpVu3TJpETWMjL730ktmhCCeTMVIO8MorrxDk68vczEynH3tMQgKHDx/m8OHDTj+2MI4rX5jQWpOdne3wbn3Q+gVxWloa3+TnO/xYQhitoKDAIWtIdSYrJYV7Zs5kyZIlvPnmm045phD9UVFRgdaaaBcopMYlJTExOZl//vOfMmbXwzi1kDJjsUNna2xs5PXXX+eCoUMJMaE5OislBYC1a9c6/dhCABw6dIiKigpG2CY/cbTpAwaQW1bGkSNHnHI8IYxSUFDg8O6v7f3szDMZn5TEDQsXsnPnTqcdV4i+KC0tBXCJFimAmydNYteuXXzyySdmhyKcyKmFlLMXOzTDxx9/TFlZGVeMHm3K8ccmJuLn7S1jRjyEFS9ObN/eOqzLGS1SANMGDADgq6++csrxhDBCfX09RUVFTi2k/Hx8+M/3v0+Ary/z58/n2DGXbPQWBlFKzVNK7VZK5Sqlft7J/dcopbbZfr5RSo1td98BpVS2UmqLUsohi/mVlZUBuESLFMD3Ro0iLjiYx374Q7NDEU4kXfsMtnTpUiICAjhn0CBTju/v48PYxEQppDyEFS9OOGvGPruxiYkE+fry5ZdfOuV4Qhhh3759tLS0MNiBi1Z3JiU8nJeuuor8/HwuvvhiampqnHp8YQ1KKW/gH8D5wAhgoVJqRIfN8oBZWusxwMPA4g73n621Hqe1znJEjPZCKiow0BG7N5y/jw+3TZnC6n372j4Hhfszcvpzy63R4WxHjhzhjTfe4MoxY/Dz8TEtjkkpKWzcuJGGhgbTYhCeKzs7m8TQUCKddBXR19ubyampfP755045nhBG2LNnDwBDnFxIQWsr7r/nz+fbb77hyiuvlM8KzzQZyNVa79daNwCvAvPbb6C1/kZrXW77cy2Q4swA7V37nPVZYoSbsrII8vXlT3/6k9mhCCcxctY+y63R4WzPPPMMjY2N3D55sqlxTE5Npa6uTqZBd2GufGFi06ZNjE5IcOoxzxk0iO3bt7etyyOE1e3evRuAQSYUUgDzR47kzxddxIcffshVV11FY2OjKXEI0yQDB9v9XWi7rSs3Ayva/a2BVUqpTUqp2zp7QH+7nrta1z5oHc913YQJvPzyy+Tl5ZkdjnAC6dpnkPr6ev75xBPMGTKEwTExpsZin3BCuve5Lle9MFFRUcGOHTuYlOLUC5fMs82Q+cEHHzj1uEL01Z49e0hISCAsIMC0GG7IyuKJCy7gnXfe4dLRo6mvrzctFuF0na1N0el0c0qps2ktpO5rd/N0rfUEWrsG/lgpdeYpO+tn1/PS0lJ8fX0J8fPr9WPNdNf06Xhrze9//3uzQxFOIIWUQZYuXUrJiRPcPmWK2aGQEh5OclgYny/u2J1ZCMfasGED8L9i3lmGxMQwMCqK9957z6nHFaKvdu/ezdChQ80Og1snT+aJCy7gw927mT9/voyZ8hyFQGq7v1OAoo4bKaXGAM8C87XWZfbbtdZFtn9LgLdo7SpoqLKyMmJiYhy+HqHRksLCuH7iRJYsWSKtUh5ACikD1NTU8OCDD5KVnGzaJBMdzczI4Iv9+2lpaTE7FOFB1q5di1KKCcmn6yFiPKUUczMz+fTTTzlx4oRTjy1EX+zevZtME9Ya7Mytkyfz10suYdWqVcydO5eKigqzQxKOtwEYopTKUEr5AQuAd9tvoJRKA94ErtNa72l3e7BSKtT+OzAH2G50gGVlZUSb1PW1v34yYwbe3t48+OCDZociHEwKKQM89dRTHDp0iN/OmWOZKyfnDBrEsdpavvvuO7NDER5k3bp1DB8+nHATuivNy8ykvr6ejz/+2OnHFqI3jh07RmlpKWkWWkj6+gkTeOGKK1i3bh2zZs2iqOiUxgnhRrTWTcAdwEpgJ/C61jpHKbVIKbXIttmvgWjgnx2mOY8HvlZKbQXWAx9orT8yOsbS0lJiTB4q0VdJYWHcfffdLF26lM2bN5sdjnAgKaT6KTc3l9///vdcfPHFnGFbz8YKzho4EIBVq1aZHInwFFprvv30U8ab1J/9jAEDiI6O5pVXXjHl+EL0lH3GPmdPfd6dS0eO5PUFC9i/axdnnHEGu3btMjsk4UBa6w+11pla60Fa60dstz2ttX7a9vstWutI2xTnbdOc22b6G2v7GWl/rNGO7tpFWHGxI3btFD9SisiAAH7605+idafDz4QbkEKqH+rq6rjyyivxaWriEYt06bOLCwlhdEICK1euNDsU4SH27dvHsdpap080Yefr7c0Vgwfz9vLlstCosDQzpz7vztmDBvH+DTdQU1rKtPHjZX02YZqymhqiXGjGvo7CAwL4+Vln8emnn7J8+XKzwxEOIoVUH+372c+4ePRotmzZwtOXXUZaRITZIZ3inEGD+Oabb6iurjY7FOEBvv76a8D5E020d824cTQ0N7Ns2TLTYhCiO7t378bHy4sBkZFmh9KpcUlJfHzLLcQGBzN79mz+85//mB2S8DBaa8pra11q6vPO3JSVxfjx47nrrruoqqoyOxzhAFJIdaOlpYWHZs/mjAEDSAkPZ9bAgcydO5fJf/87X+bl8eSFFzLPAjMvdeacQYNobGzkk08+MTsU4QFWrFhBfEgII+PjTYthTGIiYxISeOGFF0yLQYjubNu2jcHR0fh6e5sdSpfSIyP5+JZbmJKUxPXXX8/9998vkxcJp6mqqqKppYXIwECzQ+kXH29v/jhxIocPH+aBBx4wOxzhAFJIdeMPf/gDv/nkE+qampg+YAAnGho4duwYswYOZPWtt3LTpElmh9ilMwYMIDooiFdffdXsUISba2pq4uOPP+bcwYNNn3DlugkT2Lx5M2vWrDE1DiE6o7Vm/fr1Tp/Zsi8iAgNZfu213HrrrTz66KPMnz+fyspKs8MSHqC0tBRwrcV4uzIxJYXbJ0/m73//u1zYdkM+ZgdgZatXr+ZXv/wlV44ezeLLLzf9C2Jv+Xp7c+mIESx77z2OHz9OSEiI2SEJN7V+/XrKy8uZfe65ZofC1ePG8fiGDTz88MN89JHhE0kJ0S8FBQWUlJQw0cIX4drz8/Hh8aQkhpx/Pr9YsYLJkyfz1ltvMWLECLNDE26srKx1ySpXHiPV3oPnncen+/Zxww03sG3bNqKioswOSRhEWqS6oLXmnnvuYWBUFH+5+GKXK6Lsvjd6NLW1tbz77rvdbyxEH61YsQIvpTjbApOuBPv58eOxY1m5ciXr1683OxwhTrJu3ToAJrpAi5SdUorbpkzh3euuo7KykkmTJrF06VKzwxJuzN4iFeXiXfvsAn19WXz55ZQcPszCqVOlm6wbkUKqC19//TXbtm3j/51xBsEmTedshKmpqaSkpMiU0MKhVqxYweSUFCIs8qF386RJRAYG8rMFC2TaWWEp69evx9/fnxFxcWaH0mvT09P5/JprGBcby3XXXcdNN90kC2ALh7C3SLlD1z67cUlJPDpvHqv27uXhhx82OxxhECmkuvCPf/yDiIgIrhg92uxQ+sXLy4sr09P58IMPOHDggNnhCDe0Z88eNm3axPkWmnQl1N+fB84+my/y8mTGMWEp3/z3v4yJjcXPxzV71ieGhfHu9dfzfzNnsuSFF5gwYQIbNmwwOyzhZtyta5/dzZMmsWDsWH7zm9/I7LJuQgqpThQVFbF8+XJuuukmgly4NcrulkmT8FKKv/71r2aHItzQCy+8gLe3N98fO9bsUE5yU1YWU1JT+clPfsKRI0fMDkcImpqa2Hr4sEtMNHE6Pt7e/PLcc3nnBz/gxJEjTJs2jV//+tc0NDSYHZpwE0VFRfh5exPm7292KIZSSvHUxRczfcAAbrjhBj7++GOzQxL9JIVUJ5YuXUpTUxPXNDaaHYohksPDuXzkSJ599lmZcUkYqqmpiRdffJELLriAhNBQs8M5iZeXF3+95BJqq6q4dPJk6urqzA5JeLjt27dT09jIhKQks0MxxJkZGaz54Q+5YuRIHn74YSZOnMi3335rdljCDezcuZMhMTF4ebnf11R/Hx9eXrCAzKgoLrngApnJz8W53xnaT1pr/vOf/zA5JYVBFlx1vq9+NG0a1dXV/OMf/zA7FOFGVqxYweHDh1kQHGx2KJ0aGhvL05ddxrqDB7n55ptpbm42OyThwd577z0UMGvgQLNDMUxEYCDPXH45ry5cSHlBAdOnT2fRokVtkwUI0Rc7duxgaEyM2WE4TERgIO9cfz0Do6K4+OKLeeutt8wOSfSRFFIdbNu2je3bt3PVmDFmh2KocUlJnD90KI8++ihFRUVmhyPcgNaaxx57jISQEOYMGWJ2OF2aP3Ikvz73XJYtW8bChQupr683OyThod58800mp6ZarvXWCPOGDmXtj3/MD6dM4dnFixmSmsofzj9fuvuJXqupqSEvL4+hsbFmh+JQMcHBvPeDHzA6Jobvfe97/OlPf5LJkVyQFFIdLF26FB8fHy4fNcrsUAz3+7lzaWxs5L777jM7FOEG3nvvPdasWcN9Z52Fr7e32eGc1j0zZ/K7OXP473//y8zBg8nLyzM7JOFh8vLy2LJlCxcNH252KA4T6u/P7+fN4+sf/pBxSUn84qOPGD58OP/5z3+kNVj02O7du9FaM8zNCymA6OBg3rn+ei4aNoz/+7//46qrrqKiosLssEQvSCHVTkNDA0uXLmX2oEFuN1MMQEZUFHdOnszSpUt55513zA5HuLCmpiZ+dvPNDI6O5rrx480Op0fuOOMMllx5JbuPHmXcuHH89a9/pdFNxkEK67N33bl42DCTI3G84XFxWrDzRwAAD0hJREFUvHXddbxxzTWEhYVx/fXXMyw+nhdeeEFaqES3duzYAeD2LVJ2QX5+vHTVVfx29mzeeustRo8eLYvJuxAppNpZvnw5R44c4aasLLNDcZj/O/NMxiclcf3117Nnzx6zwxEu6oEHHmB3aSm/Oe88fCzeGtXepSNH8uWiRUyMieGuu+5iRGIiL774ohRUwqG01rz22muMio8nPSrK7HCcQinFeUOG8Okll/DilVcS6OvLTTfdRHp6Or///e8pKSkxO0RhUTt27MDHy4uBHpIr0Jov/2/6dD6+6SaC6+o4//zz+d73vsf+/fvNDk10Qwqpdv76178yZMgQzh00yOxQHCbA15cXr7oKX19fzj//fFlbSvTam2++yeOPP85NWVku2U0pPTKSN6+7jpcXLCDAx4cbbriBtLQ0fvWrX7F3716zwxNu6JNPPmH9+vVcP2GC2aE4nZeXF/NHjuTL22/njWuuYdSoUTzwwAOkpKRw5ZVX8uGHH8qFDHGSHTt2MCgqymXXWuuPCcnJfHH77fzynHP46KOPGDp0KLfffjv79u0zOzTRBSmkbDZs2MDatWu5ZfBgt5xus720iAhev+wyyoqKmDFjBps3bzY7JOEiXnvtNRYuXMgE2wrtrkopxYXDhvHVokW8fvXVjAkN5ZHf/Y7MzEyysrJ45JFH2Lp1qwz8Ff2mteaXv/wlaWlp/GDiRLPDMY29heq/06ez4Y47uGXiRD778EMuvPBCkpKSWLRoEStXrpTJYETrjH0e0q2vMwG+vvzfmWeyadEibhw/nheefZYhQ4Zw6aWX8uGHH9LU1GR2iKId964YeqilpYV7772X8IAAFlpsUVFHmZiSwgc33IDWmilTpvDwww9TU1NjdljCoqqqqrj77rtZsGABExMSWH7ttfi7wdVCpRRzMjN5/ZpryLnnHh6ePRuv4mJ++ctfMm7cOJKSkrj66qt55pln2L59uwyYF7322muvsX79ev5v3Di3yBkjDImJ4dF589h5zz28vGABZ8bHs/T555k3bx7R0dHMnz+fv/3tb2RnZ9PS0mJ2uMKJ6uvryc3NJdODCym7xLAwnrjwQrJ/8hPumTGDbz75hAsvvJDk5GR+/OMfs3LlSlkf0QKUWVdcs7Ky9MaNG005dkdPPfUUd999N/+89FKuHjfO7HCcqrymhns/+IA3c3KIj4/nRz/6Eddffz3p6elmh2ZpSqlNWmtLDaYzOqeam5vZunUrr732Gs8//zxlpaXclJXFI3PnEuDra9hxrOhIdTWf5Oby+b59fHXgAMXHjwMQEhLChAkTGDduHGPGjGHkyJEMGzaMiIgIcwN2ce6aT2vXruWcc85heGQkq26+2aXGEzpbbWMjX+zfz6q9e/l03z4OlJcDEBkZyZQpU5g0aRITJkxgzJgxpKenu33Pkf6yWk71NJ+ee+45brnlFv57zTXMtvCyGmZoaGpi5d69LM/OZuWePdQ2NREQEMDMmTOZOXMm06ZNY+LEiURGRpodqts5XT4ZVkgppeYBTwHewLNa6z+cbnurFFLLli3j5h/8gFkDB/LqwoUopcwOyRRrCwp4/Isv+NTWD3fUqFGcddZZTJw4kZEjR5KZmUl4eLjJUVqHMz6knJVTWmvy8/PZsmULO3bsIDc3l927d5OTk0NlZSXeSnH+0KHcM3MmE5KT+/ZkXJjWmgPl5awtKGDzoUNsPXyY7ceOndSCGxcXx6BBg8jIyCAjI4PU1FSSk5NJSkoiISGB2NhYfN28+OwPd8onaD1nli1bxp133kkEsOrmm4kNCenTvjzVgfJyvsnPZ21BARsLC9l19Cgttu8rQUFBZGZmkpmZ2ZZ3aWlppKSkkJSUREREhMd+ltv1JKe6ywnV+iI+BVwA1AA3aK039+SxHfUkn2pqahgyZAhpaWmsmDvX49/D06ltbOTLvDw+27ePL/Py2FlSgv3bfFpaGqNGjWLYsGEMGTKEgQMHkpaWRmpqKsHBwabG7aocXkgppbyBPcBsoBDYACzUWu/o6jGOLqS01jQ3N9Pc3ExTUxONjY3U1dVx4sQJiouL2bFjB2+++SYrVqxgWloaL111lXzQAQUVFbydk8On+/ax4eBBTrQbBBweHk5SUhLx8fHExMQQFRVFREQEoaGhhIaGEhISQlBQEIGBgQQGBuLv74+/vz9+fn74+fnh6+vb9uPt7Y2Pjw/e3t4n/Xh5eZ3yY8X/TB39xc8ROdXS0sLx48cpKSlh3759ZGdn8+233/LNN99w5MiRtu3iQ0IYFB3NsNhYpqalMSsjg3g3XEC0P1paWsivqGBnSQl7SkvZf+wY+48dI7+8nKKqKpo7+X81MjKS6OhooqKiiIyMJDw8nPDwcMLCwggNDSU4OJjg4OCTciggIKAth/z9/U/KIR8fn05zyJ5H3t7eKKUsnUd2rphPdo2NjRw/fpzS0lL27dvHhg0beOONN9i2bRtZyck8+73vecxMfY5U09BATnExO0tK2Hn0KHtteZdfXn5Kvvn7+xMXF9f2ORUZGUlERERbvoWEhJyUbwEBAW25Zs+3znLNnm/2/LLy51V3OdWTnFBKXQDcSWshNQV4Sms9xRH5VFBQwMMPP8yzzz7LihtvZNqAAb14tqKitpYtRUVsOXyYnOJidpSUsK+sjLoO46nCwsLaLu61z4/w8PC273L23Gj/GdQxN9rnQ1d5oZSyTD701+nyyagO25OBXK31ftsBXwXmA10m1encd999/OMf/zjldq112+Dvzn7XWtPS0tLjPtUpYWHcf/bZ3DNjhnS5sEmLiOD/TZ/O/5s+neaWFnLLylq/KJaVUVhZyZHjxzm6bx/btm+nvLaWqro66p00bsSelO2Ts/2/nf3e/rGn+7ujadOm8fHHHxsVel8YllO7d+9mzJgxna7fkh4ZyZkpKUyZOJFxiYkMjY0lxN+/v7G7PS8vLzKiosiIiuKCDvc1t7RwpLqaI9XVHK6upvj4cUpPnKD0xAmO1dZSXlbG0cJC9tXXU1VXx/GGBmqcOGtZxzzqKofab9/xto739cT69esZMWJEf0LvD0M/o+bOncvnn39OU1NTp583k1NSeOrii7l2/Hi8pQuaIYL8/JiUmsqk1NSTbm9qbuZwdTWFlZUUVVVx5PhxiqurKa2poezECcqPHuVgXR0VtbVU19dT66SB+vbcaF9c9TTXOv7e2W233XYbTz75ZH9C7ElOzAde0q1fttYqpSKUUolAeg8e22P/+c9/uP766wG4YeJEKaL6ICIwkLMGDeKsdrNOt7S0cLi6mgPl5RyqquJQZSWHq6spPXGCkkOH2Lt3L5W23DjhwM+gzj5zevudravb+rJNZ7y9vfu8ELJRLVJXAPO01rfY/r4OmKK1vqPDdrcBt9n+HArs7sdhY4DSfjze2VwtXnC9mJ0Z7wCttcNGw5qUU/1l9fNF4usfR8Yn+eRYVj+3nMWTXofT5lRPckIp9T7wB63117a/VwP30VpIeXI+9YQnnWs95cqvSZf5ZFSLVGcl4CkVmtZ6MbDYkAMqtdFKAym742rxguvF7GrxdsPpOdVfVn/9Jb7+sXp83XC5fDKSi793hpHX4SQ9yYmutvHofOoJOddO5a6viVGFVCHQvr09BSgyaN9CeCLJKSGMI/kkxMl6khNdbePXg8cK4RGM6ry9ARiilMpQSvkBC4B3Ddq3EJ5IckoI40g+CXGynuTEu8D1qtVUoFJrfbiHjxXCIxjSIqW1blJK3QGspHUqzOe11jlG7Ps0XK252NXiBdeL2dXi7ZJJOdVfVn/9Jb7+sXp8XXLRfDKSy753BpPXwaarnFBKLbLd/zTwIa0z9uXSOv35jad7rAlPw8rkXDuVW74mpi3IK4QQQgghhBCuSuZlFUIIIYQQQohekkJKCCGEEEIIIXrJ8oWUUmqeUmq3UipXKfXzTu5XSqm/2u7fppSaYEac7eLpLt6zlFKVSqkttp9fmxFnu3ieV0qVKKW2d3G/1V7f7uK11Ovrbqx+flv9fLb6+auUSlVKfaaU2qmUylFK3dXJNpb6P0F0rbvzzVP05LwWwghyrnVOKeWtlPpOta5N5lYsXUgppbyBfwDnAyOAhUqpER02Ox8YYvu5DfiXU4Nsp4fxAnyltR5n+/mtU4M81RJg3mnut8zra7OE08cL1np93YaLnN9LsPb5vARrn79NwL1a6+HAVODHVv4/V3RrCd2fb56gJ+e1EEaQc61zdwE7zQ7CESxdSAGTgVyt9X6tdQPwKjC/wzbzgZd0q7VAhFIq0dmB2vQkXkvRWn8JHDvNJlZ6fXsSr3Acy5/fVj+frX7+aq0Pa603236vpvWDL7nDZpb6P0F0zernm7P08LwWot/kXDuVUioFuBB41uxYHMHqhVQycLDd34WcekL2ZBtn6Wks05RSW5VSK5RSI50TWp9Z6fXtKVd6fV2JO5zfrnA+W+L1U0qlA+OBdR3ucoXXUIhOnea8FsJQcq61+QvwM6DF5DgcwpB1pBxIdXJbx/nae7KNs/Qkls3AAK31caXUBcDbtHaRsSorvb494Wqvrytxh/Pb6uezJV4/pVQIsBy4W2td1fHuTh5ipddQiE51c14LYRg511oppS4CSrTWm5RSZ5kcjkNYvUWqEEht93cKUNSHbZyl21i01lVa6+O23z8EfJVSMc4Lsdes9Pp2ywVfX1fiDue3pc9nK7x+SilfWr8AvKy1frOTTSz9GgrRmR6c10IYQs61k0wHLlFKHaB1OMA5Sqml5oZkLKsXUhuAIUqpDKWUH7AAeLfDNu8C19tmkpoKVGqtDzs7UJtu41VKJSillO33ybS+B2VOj7TnrPT6dssFX19X4g7nt6XPZ7NfP9uxnwN2aq2f7GIzS7+GQnTUw/NaiH6Tc+1kWutfaK1TtNbptH5n+FRrfa3JYRnK0l37tNZNSqk7gJWAN/C81jpHKbXIdv/TwIfABUAuUAPcaPF4rwB+qJRqAmqBBVpr07rFKKVeAc4CYpRShcCDgC9Y7/WFHsVrqdfXnbjC+W3189kFzt/pwHVAtlJqi+22+4G0djFa6v8E0bXOzjet9XPmRmWKTs9rW6uvEEaSc83DKPmOKYQQQgghhBC9Y/WufUIIIYQQQghhOVJICSGEEEIIIUQvSSElhBBCCCGEEL0khZQQQgghhBBC9JIUUkIIIYQQQgjRS1JICSGEEEIIIUQvSSElhBBCCCGEEL30/wHX0Fn/j3byEAAAAABJRU5ErkJggg==
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<h2 id="Closing">Closing<a class="anchor-link" href="#Closing">¶</a></h2><p>I would like to thank Lin Zheng and the commenter "Mechanical Turk" for pointing out a mistake in the original version of this article. I would also like to thank Ryan Cotterell for valuable feedback and discussions that greatly improved this article.</p>
<p>This Jupyter notebook is available for <a href="https://github.com/timvieira/blog/blob/master/content/Order-Statistics.ipynb">download</a>.</p>
<p>I am very happy to answer questions! There are many ways to do that: comment at the end of this document, Tweet at <a href="https://twitter.com/xtimv">@xtimv</a>, or email tim.f.vieira@gmail.com. If you found this article interesting please consider sharing it on social media. If you found this article useful please cite it</p>
<div class="highlight"><pre><span></span><span class="nc">@misc</span><span class="p">{</span><span class="nl">vieira2021order</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">author</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{Tim Vieira}</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">title</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{On the Distribution Function of Order Statistics}</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">year</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{2021}</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">url</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{https://timvieira.github.io/blog/post/2021/03/18/on-the-distribution-functions-of-order-statistics/}</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
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<h2 id="Appendix">Appendix<a class="anchor-link" href="#Appendix">¶</a></h2>
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<p><strong>Symbolic tests for utility methods <code>B</code> and <code>C</code></strong></p>
<p>I have included some extra material for testing the efficient algorithms for summing over subsets of size $=K$ and size $\ge K$. This strategy compares the efficient and inefficient algorithms (brute-force enumeration) using symbolic mathematics. Using symbolic mathematics ensures that the tests do not depend on any specific numerical values. Furthermore, it is helpful during debugging as we can see which terms in the output expressions are different.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">sympy</span>
<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
<span class="kn">from</span> <span class="nn">arsenal.maths.combinatorics</span> <span class="kn">import</span> <span class="n">powerset</span>
<span class="k">def</span> <span class="nf">C_slow</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">z</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span>
<span class="k">for</span> <span class="n">K</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="n">z</span><span class="p">[</span><span class="n">K</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">Y</span> <span class="ow">in</span> <span class="n">powerset</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)):</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span> <span class="o">>=</span> <span class="n">K</span><span class="p">:</span>
<span class="n">z</span><span class="p">[</span><span class="n">K</span><span class="p">]</span> <span class="o">+=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">prod</span><span class="p">([</span><span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">])</span> <span class="o">*</span> <span class="n">sympy</span><span class="o">.</span><span class="n">prod</span><span class="p">([</span><span class="n">q</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">])</span>
<span class="k">return</span> <span class="n">z</span>
<span class="k">def</span> <span class="nf">B_slow</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">z</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span>
<span class="k">for</span> <span class="n">Y</span> <span class="ow">in</span> <span class="n">powerset</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)):</span>
<span class="n">K</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
<span class="n">z</span><span class="p">[</span><span class="n">K</span><span class="p">]</span> <span class="o">+=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">prod</span><span class="p">([</span><span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">])</span> <span class="o">*</span> <span class="n">sympy</span><span class="o">.</span><span class="n">prod</span><span class="p">([</span><span class="n">q</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">])</span>
<span class="k">return</span> <span class="n">z</span>
<span class="k">def</span> <span class="nf">test_symbolic</span><span class="p">():</span>
<span class="n">Xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">Symbol</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="s1">'uvwxyz'</span><span class="p">])</span>
<span class="n">Ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="s1">'UVWXYZ'</span><span class="p">])</span>
<span class="c1"># Binomials</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Xs</span><span class="p">)):</span>
<span class="n">have</span> <span class="o">=</span> <span class="n">B</span><span class="p">(</span><span class="n">Xs</span><span class="p">,</span> <span class="n">Ys</span><span class="p">)[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
<span class="n">want</span> <span class="o">=</span> <span class="n">B_slow</span><span class="p">(</span><span class="n">Xs</span><span class="p">,</span> <span class="n">Ys</span><span class="p">)[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
<span class="k">assert</span> <span class="n">have</span> <span class="o">==</span> <span class="n">want</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'Binomials size = K: passed'</span><span class="p">)</span>
<span class="c1"># Binomials >= size K.</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Xs</span><span class="p">)):</span>
<span class="n">have</span> <span class="o">=</span> <span class="n">C</span><span class="p">(</span><span class="n">Xs</span><span class="p">,</span> <span class="n">Ys</span><span class="p">)[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
<span class="n">want</span> <span class="o">=</span> <span class="n">C_slow</span><span class="p">(</span><span class="n">Xs</span><span class="p">,</span> <span class="n">Ys</span><span class="p">)[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
<span class="k">assert</span> <span class="n">have</span> <span class="o">==</span> <span class="n">want</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'Binomials size ≥ K: passed'</span><span class="p">)</span>
<span class="n">test_symbolic</span><span class="p">()</span>
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<pre>Binomials size = K: passed
Binomials size ≥ K: passed
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<p><strong>Inspecting the symbolic output</strong></p>
<p>Below, we look at how the expression evaluated the subsets of size $=K$ in the <code>B</code> method. As a mnemonic, lower-case symbols represent the weight of <em>inclusion</em> into the set, and an upper-case symbols represent the weight of <em>exclusion</em> into the set. We consider a set with $3$ elements.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">b</span> <span class="o">=</span> <span class="n">B</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">Symbol</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="s1">'xyz'</span><span class="p">]),</span>
<span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">Symbol</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="s1">'XYZ'</span><span class="p">]))</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sympy</span><span class="o">.</span><span class="n">Matrix</span><span class="p">([</span><span class="n">b</span><span class="p">])</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
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$\displaystyle \left[\begin{matrix}X Y Z\\X Y z + X Z y + Y Z x\\X y z + Y x z + Z x y\\x y z\end{matrix}\right]$
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<p>The different set sizes are the rows of the vector above. The first row is the weight of sets of size $0$, which is $X Y Z$ because all $3$ elements are excluded. The next line is we see that each term has one lower-case factor; this is because there is one element included in the set. See the pattern?</p>
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<p><strong>Finite-difference test</strong></p>
<p>Below, we test gradients of the <code>C</code> function used in the pdf computation via the strategy described <a href="https://timvieira.github.io/blog/post/2017/04/21/how-to-test-gradient-implementations/">here</a>.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">fdcheck</span>
<span class="k">def</span> <span class="nf">test_grad</span><span class="p">():</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">12</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="n">N</span><span class="p">)</span>
<span class="k">for</span> <span class="n">K</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="p">[</span><span class="n">_</span><span class="p">,</span> <span class="n">d_p</span><span class="p">,</span> <span class="n">d_q</span><span class="p">]</span> <span class="o">=</span> <span class="n">d_C</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
<span class="n">fdcheck</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">C</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">)[</span><span class="n">K</span><span class="p">],</span> <span class="n">p</span><span class="p">,</span> <span class="n">d_p</span> <span class="o">-</span> <span class="n">d_q</span><span class="p">,</span> <span class="n">quiet</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'fdcheck passed.'</span><span class="p">)</span>
<span class="n">test_grad</span><span class="p">()</span>
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<pre>fdcheck passed.
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<p><strong>Properties of $\nabla C$</strong></p>
<p><strong>Lemma</strong>: For any $j \in N$, the following differential identities hold:
$$
\frac{\partial}{\partial p_j} C^N_K = C^{N \smallsetminus j}_{K-1}
\quad\text{and}\quad
\frac{\partial}{\partial q_j} C^N_K = C^{N \smallsetminus j}_{K}
$$</p>
<p>We use these identities in our derivation of the PDF.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">test_gradient_properties</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">ds</span><span class="p">)</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">ds</span><span class="p">)):</span>
<span class="n">ts</span> <span class="o">=</span> <span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">rvs</span><span class="p">()[</span><span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]</span>
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">:</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="n">ps</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">m</span><span class="o">.</span><span class="n">ds</span><span class="p">])</span>
<span class="p">[</span><span class="n">_</span><span class="p">,</span> <span class="n">d_p</span><span class="p">,</span> <span class="n">d_q</span><span class="p">]</span> <span class="o">=</span> <span class="n">d_C</span><span class="p">(</span><span class="n">ps</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">ps</span><span class="p">,</span> <span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">h</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">ds</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">ds</span><span class="p">);</span> <span class="n">ds</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">Ordered</span><span class="p">(</span><span class="n">ds</span><span class="p">)</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="c1"># we use this identity in fast method to evaluate the pdf.</span>
<span class="n">h</span> <span class="o">+=</span> <span class="p">(</span><span class="n">F</span><span class="p">[</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">F</span><span class="p">[</span><span class="n">k</span><span class="p">])</span> <span class="o">*</span> <span class="n">m</span><span class="o">.</span><span class="n">ds</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">F</span><span class="p">[</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">F</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">d_p</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">d_q</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">F</span><span class="p">[</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">d_p</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">F</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">d_q</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'gradient_properties: pass.'</span><span class="p">)</span>
<span class="n">test_gradient_properties</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
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<pre>gradient_properties: pass.
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</script>Animation of the inverse transform method2020-06-30T00:00:00-04:002020-06-30T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2020-06-30:/blog/post/2020/06/30/animation-of-the-inverse-transform-method/<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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<p>This is just a fun post which makes an animation of how the <a href="https://en.wikipedia.org/wiki/Inverse_transform_sampling">inverse transform method</a> works. This is a method for efficiently sampling from a univariate distribution given its quantile function. The quantile function is the inverse cumulative distribution function.</p>
<p>Let $f$ be the density function, $F$ be the cumulative density function, and $F^{-1}$ be the quantile function. For some univariate probability distribution $\mathcal{D}$.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">pylab</span> <span class="k">as</span> <span class="nn">pl</span><span class="o">,</span> <span class="nn">scipy.stats</span> <span class="k">as</span> <span class="nn">st</span>
<span class="kn">from</span> <span class="nn">arsenal.maths.rvs</span> <span class="kn">import</span> <span class="n">show_distr</span>
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<p>Below, we have a simple example distribution of a <a href="https://en.wikipedia.org/wiki/Laplace_distribution">Laplace distribution</a>.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">D</span> <span class="o">=</span> <span class="n">st</span><span class="o">.</span><span class="n">laplace</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">a</span><span class="p">,</span><span class="n">b</span> <span class="o">=</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span>
<span class="n">show_distr</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">);</span>
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<div class="prompt input_prompt">In [3]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">matplotlib.animation</span> <span class="kn">import</span> <span class="n">FuncAnimation</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">HTML</span>
<span class="n">T</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">8</span><span class="p">),</span> <span class="n">nrows</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">sharex</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="n">T</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">animate</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">clear</span><span class="p">();</span> <span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="n">u</span><span class="p">[:</span><span class="n">t</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
<span class="c1"># histogram-pdf plot</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">D</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">xs</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">25</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="nb">range</span><span class="o">=</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">))</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">t</span><span class="p">],</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">)</span>
<span class="c1"># cdf-ppf plot</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">t</span><span class="p">])</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">t</span><span class="p">],</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">D</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">xs</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">aa</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ax</span><span class="p">):</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="n">aa</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">aa</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(</span><span class="n">u</span><span class="p">[:</span><span class="n">t</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
<span class="n">aa</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([])</span>
<span class="n">aa</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([])</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">aa</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s1">'left'</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">aa</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">aa</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>
<span class="n">aa</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s1">'right'</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">aa</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s1">'top'</span><span class="p">]</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="n">aa</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_ticks_position</span><span class="p">(</span><span class="s1">'left'</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="k">return</span> <span class="p">[]</span>
<span class="n">anim</span> <span class="o">=</span> <span class="n">FuncAnimation</span><span class="p">(</span><span class="n">fig</span><span class="p">,</span> <span class="n">animate</span><span class="p">,</span> <span class="n">frames</span><span class="o">=</span><span class="n">T</span><span class="p">,</span> <span class="n">interval</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">blit</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">close</span><span class="p">(</span><span class="n">anim</span><span class="o">.</span><span class="n">_fig</span><span class="p">)</span>
<span class="n">HTML</span><span class="p">(</span><span class="n">anim</span><span class="o">.</span><span class="n">to_jshtml</span><span class="p">())</span>
</pre></div>
</div>
</div>
</div>
<div class="output_wrapper">
<div class="output">
<div class="output_area">
<div class="prompt output_prompt">Out[3]:</div>
<div class="output_html rendered_html output_subarea output_execute_result">
<link rel="stylesheet"
href="https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/
css/font-awesome.min.css">
<script language="javascript">
function isInternetExplorer() {
ua = navigator.userAgent;
/* MSIE used to detect old browsers and Trident used to newer ones*/
return ua.indexOf("MSIE ") > -1 || ua.indexOf("Trident/") > -1;
}
/* Define the Animation class */
function Animation(frames, img_id, slider_id, interval, loop_select_id){
this.img_id = img_id;
this.slider_id = slider_id;
this.loop_select_id = loop_select_id;
this.interval = interval;
this.current_frame = 0;
this.direction = 0;
this.timer = null;
this.frames = new Array(frames.length);
for (var i=0; i<frames.length; i++)
{
this.frames[i] = new Image();
this.frames[i].src = frames[i];
}
var slider = document.getElementById(this.slider_id);
slider.max = this.frames.length - 1;
if (isInternetExplorer()) {
// switch from oninput to onchange because IE <= 11 does not conform
// with W3C specification. It ignores oninput and onchange behaves
// like oninput. In contrast, Mircosoft Edge behaves correctly.
slider.setAttribute('onchange', slider.getAttribute('oninput'));
slider.setAttribute('oninput', null);
}
this.set_frame(this.current_frame);
}
Animation.prototype.get_loop_state = function(){
var button_group = document[this.loop_select_id].state;
for (var i = 0; i < button_group.length; i++) {
var button = button_group[i];
if (button.checked) {
return button.value;
}
}
return undefined;
}
Animation.prototype.set_frame = function(frame){
this.current_frame = frame;
document.getElementById(this.img_id).src =
this.frames[this.current_frame].src;
document.getElementById(this.slider_id).value = this.current_frame;
}
Animation.prototype.next_frame = function()
{
this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));
}
Animation.prototype.previous_frame = function()
{
this.set_frame(Math.max(0, this.current_frame - 1));
}
Animation.prototype.first_frame = function()
{
this.set_frame(0);
}
Animation.prototype.last_frame = function()
{
this.set_frame(this.frames.length - 1);
}
Animation.prototype.slower = function()
{
this.interval /= 0.7;
if(this.direction > 0){this.play_animation();}
else if(this.direction < 0){this.reverse_animation();}
}
Animation.prototype.faster = function()
{
this.interval *= 0.7;
if(this.direction > 0){this.play_animation();}
else if(this.direction < 0){this.reverse_animation();}
}
Animation.prototype.anim_step_forward = function()
{
this.current_frame += 1;
if(this.current_frame < this.frames.length){
this.set_frame(this.current_frame);
}else{
var loop_state = this.get_loop_state();
if(loop_state == "loop"){
this.first_frame();
}else if(loop_state == "reflect"){
this.last_frame();
this.reverse_animation();
}else{
this.pause_animation();
this.last_frame();
}
}
}
Animation.prototype.anim_step_reverse = function()
{
this.current_frame -= 1;
if(this.current_frame >= 0){
this.set_frame(this.current_frame);
}else{
var loop_state = this.get_loop_state();
if(loop_state == "loop"){
this.last_frame();
}else if(loop_state == "reflect"){
this.first_frame();
this.play_animation();
}else{
this.pause_animation();
this.first_frame();
}
}
}
Animation.prototype.pause_animation = function()
{
this.direction = 0;
if (this.timer){
clearInterval(this.timer);
this.timer = null;
}
}
Animation.prototype.play_animation = function()
{
this.pause_animation();
this.direction = 1;
var t = this;
if (!this.timer) this.timer = setInterval(function() {
t.anim_step_forward();
}, this.interval);
}
Animation.prototype.reverse_animation = function()
{
this.pause_animation();
this.direction = -1;
var t = this;
if (!this.timer) this.timer = setInterval(function() {
t.anim_step_reverse();
}, this.interval);
}
</script>
<style>
.animation {
display: inline-block;
text-align: center;
}
input[type=range].anim-slider {
width: 374px;
margin-left: auto;
margin-right: auto;
}
.anim-buttons {
margin: 8px 0px;
}
.anim-buttons button {
padding: 0;
width: 36px;
}
.anim-state label {
margin-right: 8px;
}
.anim-state input {
margin: 0;
vertical-align: middle;
}
</style>
<div class="animation">
<img id="_anim_img73bdc60a1beb42649076bc6e805a4e3b">
<div class="anim-controls">
<input id="_anim_slider73bdc60a1beb42649076bc6e805a4e3b" type="range" class="anim-slider"
name="points" min="0" max="1" step="1" value="0"
oninput="anim73bdc60a1beb42649076bc6e805a4e3b.set_frame(parseInt(this.value));"></input>
<div class="anim-buttons">
<button onclick="anim73bdc60a1beb42649076bc6e805a4e3b.slower()"><i class="fa fa-minus"></i></button>
<button onclick="anim73bdc60a1beb42649076bc6e805a4e3b.first_frame()"><i class="fa fa-fast-backward">
</i></button>
<button onclick="anim73bdc60a1beb42649076bc6e805a4e3b.previous_frame()">
<i class="fa fa-step-backward"></i></button>
<button onclick="anim73bdc60a1beb42649076bc6e805a4e3b.reverse_animation()">
<i class="fa fa-play fa-flip-horizontal"></i></button>
<button onclick="anim73bdc60a1beb42649076bc6e805a4e3b.pause_animation()"><i class="fa fa-pause">
</i></button>
<button onclick="anim73bdc60a1beb42649076bc6e805a4e3b.play_animation()"><i class="fa fa-play"></i>
</button>
<button onclick="anim73bdc60a1beb42649076bc6e805a4e3b.next_frame()"><i class="fa fa-step-forward">
</i></button>
<button onclick="anim73bdc60a1beb42649076bc6e805a4e3b.last_frame()"><i class="fa fa-fast-forward">
</i></button>
<button onclick="anim73bdc60a1beb42649076bc6e805a4e3b.faster()"><i class="fa fa-plus"></i></button>
</div>
<form action="#n" name="_anim_loop_select73bdc60a1beb42649076bc6e805a4e3b" class="anim-state">
<input type="radio" name="state" value="once" id="_anim_radio1_73bdc60a1beb42649076bc6e805a4e3b"
>
<label for="_anim_radio1_73bdc60a1beb42649076bc6e805a4e3b">Once</label>
<input type="radio" name="state" value="loop" id="_anim_radio2_73bdc60a1beb42649076bc6e805a4e3b"
checked>
<label for="_anim_radio2_73bdc60a1beb42649076bc6e805a4e3b">Loop</label>
<input type="radio" name="state" value="reflect" id="_anim_radio3_73bdc60a1beb42649076bc6e805a4e3b"
>
<label for="_anim_radio3_73bdc60a1beb42649076bc6e805a4e3b">Reflect</label>
</form>
</div>
</div>
<script language="javascript">
/* Instantiate the Animation class. */
/* The IDs given should match those used in the template above. */
(function() {
var img_id = "_anim_img73bdc60a1beb42649076bc6e805a4e3b";
var slider_id = "_anim_slider73bdc60a1beb42649076bc6e805a4e3b";
var loop_select_id = "_anim_loop_select73bdc60a1beb42649076bc6e805a4e3b";
var frames = new Array(100);
frames[0] = "data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAtAAAAJACAYAAACkMVHfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\
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<p><strong>How to read the plot:</strong> The bottom plot's y-axis ranges from $0$ to $1$. The first step of the algorithm is to sample $u \sim U(0,1)$, which we used to probe along the y-axis. The quantile function gives us fast access to the position along the x-axis for $u$. The point $x = F^{-1}(u)$ is a sample from the target distribution $\mathcal{D}$. The top plot shows empirical distribution of samples converging to $\mathcal{D}$. This is illustrated as a histogram superimposed on the probability density function $f(x)$.</p>
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</script>Generating truncated random variates2020-06-30T00:00:00-04:002020-06-30T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2020-06-30:/blog/post/2020/06/30/generating-truncated-random-variates/<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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<p>In this note, I briefly describe how to efficiently generate random variables from a truncated distribution that is sampling from $p(X \mid a < X \le b)$. The method I will describe is fairly general as it only assumes access to the cumulative distribution function (CDF) and quantile function (also known as the inverse CDF or percent-point function (PPF)).</p>
<p>Let $X \sim \mathcal{D}$ be a univariate random variable from a distribution $\mathcal{D}$ with
cumulative distribution function $F$,
probability density function $f$,
and quantile function $F^{-1}$. Below, I give an example of these functions for a normal distribution.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.stats</span> <span class="k">as</span> <span class="nn">st</span>
<span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">show_distr</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">st</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="mf">1.5</span><span class="p">)</span>
<span class="n">show_distr</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">);</span>
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<p><strong>Truncation:</strong> Suppose we'd like to sample from $\mathcal{D}$ conditioned on the value of $X$ lying in a range $a < X \le b$. This is known as a truncated random variable. Truncation, like any type of conditioning operation, may be regarded as "rejecting" a sample from $\mathcal{D}$ if it does not lie in the interval $(a,b]$. We will return to this rejection idea when we test our implementation of the truncation operation.</p>
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<p>Let $\mathcal{T}$ denote the truncated distribution, $p(X \mid a < X \le b)$</p>
<ul>
<li>Probability density function</li>
</ul>
$$
t(x) = \begin{cases}
\frac{ f(x) }{ F(b) - F(a) } & \text{if } a < x \le b \\
0 & \text{otherwise}
\end{cases}
$$<ul>
<li>Cumulative distribution function</li>
</ul>
$$
T(x) = \begin{cases}
\frac{ F(x) - F(a) }{ F(b) - F(a) } & \text{if } a < x \le b \\
1 & \text{if } x > b \\
0 & \text{otherwise}
\end{cases}
$$<ul>
<li>Quantile function</li>
</ul>
$$
T^{-1}(u) = F^{-1}(F(a) + u \cdot (F(b) - F(a)))
$$
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<div class="prompt input_prompt">In [2]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.maths.rvs</span> <span class="kn">import</span> <span class="n">TruncatedDistribution</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">TruncatedDistribution</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mf">.1</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">show_distr</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">);</span>
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<p><strong>Sampling:</strong> The <a href="https://en.wikipedia.org/wiki/Inverse_transform_sampling">inverse transform method</a> says that I can generate from $\mathcal{D}$ by transforming a uniform random variate $U \sim \mathcal{U}(0,1)$,</p>
$$F^{-1}(U) \sim \mathcal{D}$$<p>We can sample from our truncated distribution by the exact same mechanism.</p>
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<h2 id="Implementation">Implementation<a class="anchor-link" href="#Implementation">¶</a></h2>
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<p>Below, is a simple implementation of these ideas which is available in <a href="https://github.com/timvieira/arsenal">arsenal.maths.rvs.TruncatedDistribution</a>. The implementation is meant to work with the distributions in <code>scipy.stats</code>. Note the <code>ppf</code> is the quantile function, and <code>rvs</code> is a method to sample from the distribution.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.nb</span> <span class="kn">import</span> <span class="n">psource</span>
<span class="kn">from</span> <span class="nn">arsenal.maths.rvs</span> <span class="kn">import</span> <span class="n">TruncatedDistribution</span>
<span class="n">psource</span><span class="p">(</span><span class="n">TruncatedDistribution</span><span class="p">)</span>
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<div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">TruncatedDistribution</span><span class="p">:</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">a</span> <span class="o"><=</span> <span class="n">b</span><span class="p">),</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">d</span> <span class="o">=</span> <span class="n">d</span><span class="p">;</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="p">;</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">b</span>
<span class="bp">self</span><span class="o">.</span><span class="n">cdf_b</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">cdf_a</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">cdf_w</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cdf_b</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">cdf_a</span>
<span class="k">def</span> <span class="nf">sf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">pdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o"><</span> <span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">x</span> <span class="o"><=</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">d</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">cdf_w</span>
<span class="k">def</span> <span class="nf">rvs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">ppf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">d</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cdf_a</span> <span class="o">+</span> <span class="n">u</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">cdf_w</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">minimum</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o"><</span> <span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">cdf_a</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">cdf_w</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">mean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="c1"># The truncated mean is unfortunately not analytical</span>
<span class="k">return</span> <span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
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<p>We compare our inverse transform method from truncated generation to an inefficient rejection sampling approach which repeatedly samples from $\mathcal{D}$ until the sample lies the interval $[a,b]$.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">truncated_generator_slow</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">max_rejections</span><span class="o">=</span><span class="mi">10000</span><span class="p">):</span>
<span class="s2">"Sample X ~ D( X | a ≤ X ≤ b) with rejection sampling."</span>
<span class="k">assert</span> <span class="n">a</span> <span class="o"><</span> <span class="n">b</span><span class="p">,</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">]</span>
<span class="n">rejections</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">rvs</span><span class="p">()</span>
<span class="k">if</span> <span class="n">a</span> <span class="o"><=</span> <span class="n">x</span> <span class="o"><=</span> <span class="n">b</span><span class="p">:</span>
<span class="k">return</span> <span class="n">x</span>
<span class="n">rejections</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">rejections</span> <span class="o">></span> <span class="n">max_rejections</span><span class="p">:</span>
<span class="k">assert</span> <span class="kc">False</span><span class="p">,</span> <span class="s1">'too many rejections'</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal</span> <span class="kn">import</span> <span class="n">iterview</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">compare_to_rejection</span><span class="p">(</span>
<span class="n">D</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span>
<span class="n">m</span><span class="p">:</span> <span class="s1">'number of samples'</span> <span class="o">=</span> <span class="mi">100_000</span><span class="p">,</span>
<span class="n">B</span><span class="p">:</span> <span class="s1">'number of histogram bins'</span> <span class="o">=</span> <span class="mi">50</span><span class="p">,</span>
<span class="p">):</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">TruncatedDistribution</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="n">samples</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="n">B</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'histogram'</span><span class="p">)</span>
<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">truncated_generator_slow</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">iterview</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">))]</span>
<span class="n">pl</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="n">B</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'b'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'rejection histogram'</span><span class="p">)</span>
<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">T</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">xs</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'pdf'</span><span class="p">)</span>
<span class="c1"># show truncation lines</span>
<span class="n">pl</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'bounds'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span> <span class="o">*</span> <span class="mf">1.2</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'truncated distribution'</span><span class="p">);</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">compare_to_rejection</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">T</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="n">T</span><span class="o">.</span><span class="n">b</span><span class="p">)</span>
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<pre>100.0% (100000/100000) [==============================================] 00:00:04
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<p>Let's test out the generality of our approach on a Gumbel distribution.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">compare_to_rejection</span><span class="p">(</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">st</span><span class="o">.</span><span class="n">gumbel_r</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">)),</span>
<span class="n">a</span> <span class="o">=</span> <span class="mf">0.75</span><span class="p">,</span>
<span class="n">b</span> <span class="o">=</span> <span class="mf">2.5</span>
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<pre>100.0% (100000/100000) [==============================================] 00:00:10
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<h3 id="Testing-the-analytical-forms-for-the-pdf,-cdf,-and-ppf">Testing the analytical forms for the pdf, cdf, and ppf<a class="anchor-link" href="#Testing-the-analytical-forms-for-the-pdf,-cdf,-and-ppf">¶</a></h3><p>The test cases below show that the empirical distribution of the rejection sampler matches the analytical versions of the pdf, cdf, and ppf that we implemented.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">Empirical</span><span class="p">,</span> <span class="n">compare_samples_to_distr</span>
<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">truncated_generator_slow</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">T</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="n">T</span><span class="o">.</span><span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">iterview</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">100_000</span><span class="p">))]</span>
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<pre>100.0% (100000/100000) [==============================================] 00:00:04
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<div class="prompt input_prompt">In [10]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">compare_samples_to_distr</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">samples</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">);</span>
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<p>We can also check that the fast sampler matches in the same ways as above.</p>
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<div class="prompt input_prompt">In [11]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">compare_samples_to_distr</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">T</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">100_000</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">);</span>
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</script>Algorithms for sampling without replacement2019-09-16T00:00:00-04:002019-09-16T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2019-09-16:/blog/post/2019/09/16/algorithms-for-sampling-without-replacement/<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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<p>In this notebook, we'll describe, implement, and test some simple and efficient strategies for sampling without replacement from a categorical distribution.</p>
<p>Given a set of items indexed by $1, \ldots, n$ and weights $w_1, \ldots, w_n$, we want to sample $0 < k \le n$ elements without replacement from the set.</p>
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<h2 id="Theory">Theory<a class="anchor-link" href="#Theory">¶</a></h2><p>The probability of the sampling without replacement scheme can be computed analytically. Let $z$ be an ordered sample without replacement from the indices $\{1, \ldots, n\}$ of size $0 < k \le n$. Borrowing Python notation, let $z_{:t}$ denote the indices up to, but not including, $t$. The probability of $z$ is
$$
\mathrm{Pr}(z) = \prod_{t=1}^{k} p(z_t \mid z_{:t})
\quad\text{ where }\quad
p(z_t \mid z_{:t}) = \frac{ w_{z_t} }{ W_t(z) } \quad\text{ and }\quad
W_t(z) = \sum_{i=1}^n w_{i} - \sum_{i = 1}^t w_{z_i}
$$</p>
<p>Note that $w_{z_t}$ is the weight of the $t^{\text{th}}$ item sampled in $z$ and $W_t(z)$ is the normalizing constant at time $t$.</p>
<p>This probability is evaluated by <code>p_perm</code> (below), and it can be used to test that $z$ is sampled according to the correct sampling without replacement process.</p>
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<div class="prompt input_prompt">In [1]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">p_perm</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
<span class="s2">"The probability of a permutation `z` under the sampling without replacement scheme."</span>
<span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o"><=</span> <span class="nb">len</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">W</span> <span class="o">=</span> <span class="n">w</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">z</span><span class="p">)):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">w</span><span class="p">[</span><span class="n">z</span><span class="p">[</span><span class="n">t</span><span class="p">]]</span>
<span class="n">p</span> <span class="o">*=</span> <span class="n">x</span> <span class="o">/</span> <span class="n">W</span>
<span class="n">W</span> <span class="o">-=</span> <span class="n">x</span>
<span class="k">return</span> <span class="n">p</span>
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<h2 id="Algorithms">Algorithms<a class="anchor-link" href="#Algorithms">¶</a></h2>
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<h3 id="Baseline:-The-numpy-implementation">Baseline: The numpy implementation<a class="anchor-link" href="#Baseline:-The-numpy-implementation">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">swor_numpy</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">w</span> <span class="o">/</span> <span class="n">w</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="c1"># must normalize `w` first, unlike Gumbel version</span>
<span class="n">U</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="n">p</span><span class="p">,</span> <span class="n">replace</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">R</span><span class="p">)])</span>
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<h3 id="Heap-based-sampling">Heap-based sampling<a class="anchor-link" href="#Heap-based-sampling">¶</a></h3><p>Using <a href="http://timvieira.github.io/blog/post/2016/11/21/heaps-for-incremental-computation/">heap sampling</a>, we can do the computation in $\mathcal{O}(N + K \log N)$. It's possible that shrinking the heap rather than leaving it size $n$ could yield an improvement. The <a href="https://github.com/timvieira/arsenal/blob/master/arsenal/datastructures/heap/sumheap.pyx">implementation</a> that I am using is from my Python <a href="https://github.com/timvieira/arsenal/">arsenal</a>.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.datastructures.heap.sumheap</span> <span class="kn">import</span> <span class="n">SumHeap</span>
<span class="k">def</span> <span class="nf">swor_heap</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">R</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">R</span><span class="p">):</span>
<span class="n">z</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="n">SumHeap</span><span class="p">(</span><span class="n">w</span><span class="p">)</span><span class="o">.</span><span class="n">swor</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="k">return</span> <span class="n">z</span>
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<h3 id="The-Gumbel-sort-trick">The Gumbel-sort trick<a class="anchor-link" href="#The-Gumbel-sort-trick">¶</a></h3><p>The running time for the Gumbel version is $\mathcal{O}(N + N \log K)$ assuming that we use a bounded heap of size $K$. The implementation does not include the bounded heap optimization. My experiments will use $k=n$.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">swor_gumbel</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">gumbel</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">R</span><span class="p">,</span><span class="n">n</span><span class="p">))</span>
<span class="n">G</span> <span class="o">+=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">G</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
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<h3 id="Exponential-sort-trick">Exponential-sort trick<a class="anchor-link" href="#Exponential-sort-trick">¶</a></h3>
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<p>Efraimidis and Spirakis (2006)'s algorithm, modified slightly to use Exponential random variates for aesthetic reasons. The Gumbel-sort and Exponential-sort algorithms are very tightly connected as I have discussed in a <a href="http://timvieira.github.io/blog/post/2014/08/01/gumbel-max-trick-and-weighted-reservoir-sampling/">2014 article</a> and can be seen in the similarity of the code for the two methods.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">swor_exp</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">E</span> <span class="o">=</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">R</span><span class="p">,</span><span class="n">n</span><span class="p">)))</span>
<span class="n">E</span> <span class="o">/=</span> <span class="n">w</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">E</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
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<h2 id="Test-cases">Test cases<a class="anchor-link" href="#Test-cases">¶</a></h2>
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<h3 id="Correctness">Correctness<a class="anchor-link" href="#Correctness">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span><span class="o">,</span> <span class="nn">pylab</span> <span class="k">as</span> <span class="nn">pl</span>
<span class="kn">from</span> <span class="nn">numpy.random</span> <span class="kn">import</span> <span class="n">uniform</span>
<span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">compare</span><span class="p">,</span> <span class="n">random_dist</span>
<span class="n">R</span> <span class="o">=</span> <span class="mi">50_000</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">random_dist</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">methods</span> <span class="o">=</span> <span class="p">[</span>
<span class="n">swor_numpy</span><span class="p">,</span>
<span class="n">swor_gumbel</span><span class="p">,</span>
<span class="n">swor_heap</span><span class="p">,</span>
<span class="n">swor_exp</span><span class="p">,</span>
<span class="p">]</span>
<span class="n">S</span> <span class="o">=</span> <span class="p">{</span><span class="n">f</span><span class="o">.</span><span class="vm">__name__</span><span class="p">:</span> <span class="n">f</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">methods</span><span class="p">}</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">Counter</span>
<span class="kn">from</span> <span class="nn">arsenal.maths.combinatorics</span> <span class="kn">import</span> <span class="n">permute</span>
<span class="k">def</span> <span class="nf">counts</span><span class="p">(</span><span class="n">S</span><span class="p">):</span>
<span class="s2">"empirical distribution over z"</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">Counter</span><span class="p">()</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">S</span><span class="p">:</span>
<span class="n">c</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">s</span><span class="p">)]</span> <span class="o">+=</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">m</span>
<span class="k">return</span> <span class="n">c</span>
<span class="n">D</span> <span class="o">=</span> <span class="p">{</span><span class="n">name</span><span class="p">:</span> <span class="n">counts</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">name</span><span class="p">])</span> <span class="k">for</span> <span class="n">name</span> <span class="ow">in</span> <span class="n">S</span><span class="p">}</span>
<span class="n">R</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="n">permute</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)):</span>
<span class="n">R</span><span class="p">[</span><span class="n">z</span><span class="p">]</span> <span class="o">=</span> <span class="n">p_perm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
<span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">D</span><span class="o">.</span><span class="n">values</span><span class="p">():</span>
<span class="n">d</span><span class="p">[</span><span class="n">z</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">0</span>
<span class="c1"># Check that p_perm sums to one.</span>
<span class="n">np</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">assert_allclose</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">values</span><span class="p">()),</span> <span class="mi">1</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">D</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="n">compare</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span><span class="o">.</span><span class="n">show</span><span class="p">(</span><span class="n">title</span><span class="o">=</span><span class="n">name</span><span class="p">);</span>
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<pre>
Comparison: n=120
norms: <span class="ansi-green-fg">[0.150794, 0.151828]</span>
pearson: <span class="ansi-red-fg">0.999164</span>
spearman: <span class="ansi-red-fg">0.994145</span>
ℓ∞: <span class="ansi-yellow-fg">0.00251286</span>
ℓ₂: <span class="ansi-yellow-fg">0.00510196</span>
same-sign: <span class="ansi-green-fg">100.00% (120/120)</span>
regression: <span class="ansi-yellow-fg">[0.988 0.000] R=0.005</span>
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Comparison: n=120
norms: <span class="ansi-green-fg">[0.150794, 0.151176]</span>
pearson: <span class="ansi-red-fg">0.999236</span>
spearman: <span class="ansi-red-fg">0.996102</span>
ℓ∞: <span class="ansi-yellow-fg">0.00220764</span>
ℓ₂: <span class="ansi-yellow-fg">0.00472414</span>
same-sign: <span class="ansi-green-fg">100.00% (120/120)</span>
regression: <span class="ansi-yellow-fg">[0.995 0.000] R=0.005</span>
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
"
>
</div>
</div>
<div class="output_area">
<div class="prompt"></div>
<div class="output_subarea output_stream output_stdout output_text">
<pre>
Comparison: n=120
norms: <span class="ansi-green-fg">[0.150794, 0.150319]</span>
pearson: <span class="ansi-red-fg">0.999361</span>
spearman: <span class="ansi-red-fg">0.995895</span>
ℓ∞: <span class="ansi-yellow-fg">0.00150088</span>
ℓ₂: <span class="ansi-yellow-fg">0.00432134</span>
same-sign: <span class="ansi-green-fg">100.00% (120/120)</span>
regression: <span class="ansi-yellow-fg">[1.004 -0.000] R=0.004</span>
</pre>
</div>
</div>
<div class="output_area">
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<img 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<pre>
Comparison: n=120
norms: <span class="ansi-green-fg">[0.150794, 0.150936]</span>
pearson: <span class="ansi-red-fg">0.999362</span>
spearman: <span class="ansi-red-fg">0.995963</span>
ℓ∞: <span class="ansi-yellow-fg">0.00127402</span>
ℓ₂: <span class="ansi-yellow-fg">0.00429467</span>
same-sign: <span class="ansi-green-fg">100.00% (120/120)</span>
regression: <span class="ansi-yellow-fg">[0.998 0.000] R=0.004</span>
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<h3 id="Efficiency">Efficiency<a class="anchor-link" href="#Efficiency">¶</a></h3>
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<div class="prompt input_prompt">In [9]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.timer</span> <span class="kn">import</span> <span class="n">timers</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">timers</span><span class="p">()</span>
<span class="n">R</span> <span class="o">=</span> <span class="mi">50</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">18</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="n">i</span>
<span class="c1">#print('n=', n, 'i=', i)</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">R</span><span class="p">):</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">random_dist</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">methods</span><span class="p">)</span>
<span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">methods</span><span class="p">:</span>
<span class="n">name</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="vm">__name__</span>
<span class="k">with</span> <span class="n">T</span><span class="p">[</span><span class="n">name</span><span class="p">](</span><span class="n">n</span> <span class="o">=</span> <span class="n">n</span><span class="p">):</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">S</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="c1"># some sort of sanity check</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'done'</span><span class="p">)</span>
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<pre>done
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">T</span><span class="o">.</span><span class="n">plot_feature</span><span class="p">(</span><span class="s1">'n'</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">T</span><span class="o">.</span><span class="n">plot_feature</span><span class="p">(</span><span class="s1">'n'</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]);</span> <span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_yscale</span><span class="p">(</span><span class="s1">'log'</span><span class="p">);</span> <span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xscale</span><span class="p">(</span><span class="s1">'log'</span><span class="p">);</span>
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<div class="prompt input_prompt">In [11]:</div>
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<pre>swor_exp is 1.2260x faster than swor_gumbel (<span class="ansi-green-fg">p=0.00083</span>, mean: swor_gumbel: 0.00184231, swor_exp: 0.00150269)
swor_exp is 2.2546x faster than swor_heap (<span class="ansi-green-fg">p=0.00007</span>, mean: swor_heap: 0.00338799, swor_exp: 0.00150269)
swor_exp is 8.2761x faster than swor_numpy (<span class="ansi-green-fg">p=0.00000</span>, mean: swor_numpy: 0.0124364, swor_exp: 0.00150269)
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<p><strong>Remarks:</strong></p>
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<li><p>The numpy version is not very competitive. That's because it's uses a less efficient base algorithm that is not optimized for sampling without replacement.</p>
</li>
<li><p>The heap-based implementation is pretty fast. It has the best asymptotic complexity if the sample size is less then $n$.</p>
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<li><p>That said, the heap-based sampler is harder to implement than the Exp and Gumbel algorithm, and harder to vectorize, unlike Exp and Gumbel.</p>
</li>
<li><p>The difference between the Exp and Gumbel tricks is just that the Gumbel trick takes does a few more floating-point operations. In fact, as I pointed out in a <a href="http://timvieira.github.io/blog/post/2014/08/01/gumbel-max-trick-and-weighted-reservoir-sampling/">2014 article</a>, the Exp-sort and Gumbel-sort tricks produced <em>precisely</em> the same sample if we use the same random seed.</p>
</li>
<li><p>I suspect that the performance of both the Exp and Gumbel methods could be improved with a bit of implementation effort. For example, currently, there are some unnecessary extra temporary memory allocations. These algorithms are also trivial to parallelize. The real bottleneck is the random variate generation time.</p>
</li>
</ul>
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<p><strong>How to cite this article</strong>: If you found this article useful, please cite it as</p>
<pre><code>@misc{vieira2019swor-algs,
title = {Algorithms for sampling without replacement},
author = {Tim Vieira},
url = "https://timvieira.github.io/blog/post/2019/09/16/algorithms-for-sampling-without-replacement/",
year = {2019}
}</code></pre>
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</script>The restart acceleration trick: A cure for the heavy tail of wasted time2019-09-06T00:00:00-04:002019-09-06T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2019-09-06:/blog/post/2019/09/06/the-restart-acceleration-trick-a-cure-for-the-heavy-tail-of-wasted-time/<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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<p>Ever wondered why restarting stuff, possibly multiple times, seems to fix problems? For example, downloading a webpage or running an optimization algorithm from random initializations. Today, we'll get into the math behind why random restarts often accelerate the time until completion. Our discussion will focus on restart acceleration for speeding up computer algorithms, but the concepts extend to other restartable processes.</p>
<p>A <a href="https://en.wikipedia.org/wiki/Las_Vegas_algorithm">Las Vegas algorithm</a> is an algorithm with an <em>uncertain</em> runtime (i.e., time until completion), but <em>certain</em> correctness. Thus, it is "dual" to the more familiar <a href="https://en.wikipedia.org/wiki/Monte_Carlo_algorithm">Monte Carlo</a> family of algorithms, which have <em>certain</em> runtime, but <em>uncertain</em> correctness. Put differently, when a Las Vegas algorithm halts, its output will be correct, but the time until it halts (runtime) is a random variable. Thus, the algorithm has a runtime <em>distribution</em>.</p>
<p>In its most general form, the restart acceleration trick eliminates heavy tails of the runtime distribution. If the distribution is not <a href="https://en.wikipedia.org/wiki/Heavy-tailed_distribution">heavy-tailed</a> (or <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.113.8427&rep=rep1&type=pdf">bounded heavy-tailed</a>), then this trick will generally not accelerate the algorithm.</p>
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<p><strong>The random restart wrapper:</strong> The generic strategy for accelerating Las Vegas algorithms will determine a time-limiting policy, which will improve the overall algorithm's runtime distribution. To start, we will assume that the entire runtime distribution is known.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal</span> <span class="kn">import</span> <span class="n">timelimit</span><span class="p">,</span> <span class="n">Timeout</span>
<span class="k">def</span> <span class="nf">restart_acceleration</span><span class="p">(</span><span class="n">alg</span><span class="p">,</span> <span class="n">policy</span><span class="p">):</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span> <span class="c1"># repeat until solved...</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">with</span> <span class="n">timelimit</span><span class="p">(</span><span class="n">policy</span><span class="p">()):</span> <span class="c1"># Run `alg` for at most `policy()` seconds</span>
<span class="k">return</span> <span class="n">alg</span><span class="p">()</span> <span class="c1"># if the alg halted within timelimit, output result.</span>
<span class="k">except</span> <span class="n">Timeout</span><span class="p">:</span>
<span class="k">pass</span>
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<p><strong>How it works</strong>: If the algorithm halted within the chosen time limit, the wrapper also halts with the algorithm's result. Otherwise, the wrapper runs the algorithm again, hoping to get luckier.</p>
<p>Some surprising results:</p>
<ul>
<li><p>When the runtime distribution is known, the optimal policy for runtime budgets is a fixed constant threshold! This is somewhat surprising - I will give a simple argument for why it is true.</p>
</li>
<li><p>When the runtime distribution is not known, there exists a universal strategy, which retries under a universal sequence of runtime thresholds.</p>
</li>
</ul>
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<p>The seminal paper on this topic is</p>
<ul>
<li>Luby, Sinclair, and Zuckerman. 1993. <a href="http://http.icsi.berkeley.edu/ftp/global/pub/techreports/1993/tr-93-010.pdf">Optimal Speedup of Las Vegas Algorithms</a>.</li>
</ul>
<p>Lots of details (e.g., CDF, higher-order moments, optimality conditions) are worked out in the following paper. They also account for constant restart overhead, which is a very nice detail since in practice restarts are not free.</p>
<ul>
<li>van Moorsel and Katinka Wolter. 2004. <a href="https://dl.acm.org/citation.cfm?id=1026089">Analysis and Algorithms for Restart</a>.</li>
</ul>
<p>I will draw heavily on both this papers in this notebook.</p>
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<h2 id="Simulation">Simulation<a class="anchor-link" href="#Simulation">¶</a></h2>
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<p>Let's set up some simple heavy-tailed distributions to play with. (A quick reference for some heavy-tailed distributions:
<a href="https://en.wikipedia.org/wiki/Pareto_distribution">Pareto</a>, <a href="https://en.wikipedia.org/wiki/Log-normal_distribution">Log-normal</a>, <a href="https://en.wikipedia.org/wiki/Weibull_distribution">Weibull</a>, <a href="https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution">Fréchet</a>.)</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span><span class="o">,</span> <span class="nn">pylab</span> <span class="k">as</span> <span class="nn">pl</span><span class="o">,</span> <span class="nn">scipy.stats</span> <span class="k">as</span> <span class="nn">st</span>
<span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">mean_confidence_interval</span>
<span class="kn">from</span> <span class="nn">scipy.integrate</span> <span class="kn">import</span> <span class="n">quad</span>
<span class="kn">from</span> <span class="nn">arsenal</span> <span class="kn">import</span> <span class="n">iterview</span>
<span class="c1"># define a runtime distribution (uncomment to change the distribution we use throughout the notebook)</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">st</span><span class="o">.</span><span class="n">lognorm</span><span class="p">(</span><span class="mf">2.</span><span class="p">)</span>
<span class="c1">#d = st.weibull_min(.5) # heavy-tailed when parameter < 1</span>
<span class="c1">#d = st.pareto(1.1) # variance is infinite for < 2</span>
<span class="n">tmin</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="mf">.01</span><span class="p">)</span>
<span class="n">tmax</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="mf">.9</span><span class="p">)</span>
<span class="n">overhead</span> <span class="o">=</span> <span class="mf">.2</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'average runtime: </span><span class="si">{</span><span class="n">d</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s1">g</span><span class="si">}</span><span class="s1">, variance: </span><span class="si">{</span><span class="n">d</span><span class="o">.</span><span class="n">var</span><span class="p">()</span><span class="si">:</span><span class="s1">g</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
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<pre>average runtime: 7.38906, variance: 2926.36
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<h3 id="Expected-capped-running-time">Expected capped running time<a class="anchor-link" href="#Expected-capped-running-time">¶</a></h3><p>The following is recursive definition of the expected cost of a policy $\tau$ with a fixed restart-overhead penalty $\omega \ge 0$. We require $\tau \ge \min\{ t \mid p(t) > 0 \}$ to avoid dividing by zero.</p>
$$
T =
\underbrace{p(t \le \tau)}_{\text{prob of done}}
\cdot
\underbrace{\mathbb{E}_p[ t \mid t \le \tau ]}_{\text{cost given done}}
+
\underbrace{(1 - p( t \le \tau))}_{\text{prob of retry}}
\cdot
\underbrace{(\overbrace{\tau}^{\text{fixed cost}}
+
\overbrace{\omega}^{\text{fixed overhead cost}}
+
\overbrace{T}^{\text{recursion}})}_{\text{cost given restart}}
$$
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<p>Solving the linear recurrence for $T$, we obtain</p>
$$
\begin{align}
T &=
\underbrace{\mathbb{E}_p[t |t \le \tau]}_{\text{cost given no restart}}
+
\underbrace{\frac{1-p(t \le \tau)}{p(t \le \tau)}}_{\text{restart rate}}
\cdot
\underbrace{(\omega + \tau)}_{\text{cost of each restart}}.
\end{align}
$$<p>Remarks:</p>
<ul>
<li><p>This decision problem can be formalized as a simple two-state <a href="https://en.wikipedia.org/wiki/Markov_decision_process">Markov decision process</a> (MDP) (a terminal and non-terminal state) with actions corresponding to the threshold values.</p>
</li>
<li><p>Although the analysis below does not directly show this. We know that all MDPs have an optimal policy that is pure and stationary (i.e., deterministic and fixed over time). Thus, the optimal policy in our setting is a fixed threshold $\tau^*$.</p>
</li>
<li><p>The expected value isn't the only reasonable choice of what to optimize. It is possible to minimize higher-order moments as well as the expected value. It is also possible to minimize quantiles and the probability of completion by a budget.</p>
</li>
</ul>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">T</span><span class="p">(</span><span class="n">τ</span><span class="p">,</span> <span class="n">ω</span><span class="p">):</span>
<span class="s2">"Expected runtime with restart threshold τ and overhead ω."</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">τ</span><span class="p">)</span>
<span class="k">return</span> <span class="n">conditional_mean</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">τ</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="n">p</span> <span class="o">*</span> <span class="p">(</span><span class="n">τ</span> <span class="o">+</span> <span class="n">ω</span><span class="p">)</span>
<span class="nd">@np</span><span class="o">.</span><span class="n">vectorize</span>
<span class="k">def</span> <span class="nf">conditional_mean</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
<span class="s2">"E[T | a <= T < b]"</span>
<span class="k">return</span> <span class="n">d</span><span class="o">.</span><span class="n">expect</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">conditional</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">tmin</span><span class="p">,</span> <span class="n">tmax</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">Ts</span> <span class="o">=</span> <span class="n">T</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">overhead</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">Ts</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'T(R)'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">ts</span><span class="p">[</span><span class="n">Ts</span><span class="o">.</span><span class="n">argmin</span><span class="p">()],</span> <span class="p">[</span><span class="n">Ts</span><span class="o">.</span><span class="n">min</span><span class="p">()],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">75</span><span class="p">)</span>
<span class="n">rstar</span> <span class="o">=</span> <span class="n">ts</span><span class="p">[</span><span class="n">Ts</span><span class="o">.</span><span class="n">argmin</span><span class="p">()]</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'speedup: $\frac</span><span class="si">{E[T]}{E[T*]}</span><span class="s1">$ = </span><span class="si">%.4g</span><span class="s1">x faster'</span> <span class="o">%</span> <span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">/</span><span class="n">Ts</span><span class="o">.</span><span class="n">min</span><span class="p">()))</span>
<span class="n">pl</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$\mu = T(\infty)$'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="n">ts</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">ts</span><span class="o">.</span><span class="n">max</span><span class="p">())</span> <span class="p">;</span> <span class="n">pl</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">d</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">*</span><span class="mi">2</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">ts</span><span class="o">.</span><span class="n">max</span><span class="p">());</span> <span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'R'</span><span class="p">);</span> <span class="c1">#pl.xscale('log'); pl.yscale('log');</span>
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<h3 id="Unknown-runtime-distribution">Unknown runtime distribution<a class="anchor-link" href="#Unknown-runtime-distribution">¶</a></h3><p>When the runtime distribution is not known, there are still strategies which can provide acceleration. Although, now there is an important constant factor, which might result in slower performance.</p>
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<p><strong>Luby's universal strategy:</strong> The expected runtime of the universal strategy is bounded by $\mathcal{O}(T(\tau^*) \log T(\tau^*))$.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">_luby</span><span class="p">():</span>
<span class="k">def</span> <span class="nf">h</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
<span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">yield</span> <span class="mi">1</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
<span class="k">yield from</span> <span class="n">h</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
<span class="k">yield</span> <span class="mi">2</span> <span class="o">**</span> <span class="n">k</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="k">yield from</span> <span class="n">h</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">universal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">base</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
<span class="s2">"Generate the universal threshold sequence"</span>
<span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">_luby</span><span class="p">():</span>
<span class="k">if</span> <span class="n">base</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
<span class="k">yield</span> <span class="n">scale</span> <span class="o">*</span> <span class="n">base</span> <span class="o">**</span> <span class="n">np</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">yield</span> <span class="n">scale</span> <span class="o">*</span> <span class="n">x</span>
<span class="kn">from</span> <span class="nn">arsenal.iterextras</span> <span class="kn">import</span> <span class="n">take</span>
<span class="k">assert</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span>
<span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span>
<span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span>
<span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span>
<span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="nb">list</span><span class="p">(</span><span class="n">take</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="n">_luby</span><span class="p">()))</span>
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<p><strong>Geometric sequence:</strong> An alternative sequence, which often works better in practice, is a gemetric sequence. Note, however, that this sequence can suffer <em>unbounded</em> regret in the worst case, unlike the universal sequence.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">geometric</span><span class="p">(</span><span class="n">scale</span><span class="p">,</span> <span class="n">base</span><span class="o">=</span><span class="mf">2.0</span><span class="p">):</span>
<span class="s2">"Generate the geometrically increasing threshold sequence"</span>
<span class="k">assert</span> <span class="n">base</span> <span class="o">></span> <span class="mi">1</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">yield</span> <span class="n">scale</span> <span class="o">*</span> <span class="n">base</span><span class="o">**</span><span class="n">i</span>
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<p>Let's compare the strategies.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal</span> <span class="kn">import</span> <span class="n">restore_random_state</span>
<span class="k">def</span> <span class="nf">_simulate</span><span class="p">(</span><span class="n">policy</span><span class="p">,</span> <span class="n">seed</span><span class="p">):</span>
<span class="k">with</span> <span class="n">restore_random_state</span><span class="p">(</span><span class="n">seed</span><span class="p">):</span>
<span class="n">total</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">r</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">rvs</span><span class="p">()</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">policy</span><span class="p">()</span> <span class="c1"># this version uses a callable policy</span>
<span class="n">total</span> <span class="o">+=</span> <span class="nb">min</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
<span class="k">if</span> <span class="n">r</span> <span class="o"><=</span> <span class="n">R</span><span class="p">:</span>
<span class="k">return</span> <span class="n">total</span>
<span class="n">total</span> <span class="o">+=</span> <span class="n">overhead</span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">seeds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">**</span><span class="mi">32</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">M</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span> <span class="c1"># use common random numbers</span>
<span class="n">U</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">_simulate</span><span class="p">(</span><span class="n">universal</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="fm">__next__</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seeds</span><span class="p">])</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">_simulate</span><span class="p">(</span><span class="n">geometric</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="fm">__next__</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seeds</span><span class="p">])</span>
<span class="n">O</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">_simulate</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">rstar</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seeds</span><span class="p">])</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
<span class="c1"># First and second moments</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'uncapped: </span><span class="si">{</span><span class="n">d</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1"> </span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">D</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'universal: </span><span class="si">{</span><span class="n">U</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1"> </span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">U</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'geometric: </span><span class="si">{</span><span class="n">G</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1"> </span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">G</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'optimized: </span><span class="si">{</span><span class="n">O</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1"> </span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">O</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
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<pre>uncapped: 7.39 1081.91
universal: 1.59 5.99
geometric: 2.04 12.44
optimized: 1.42 4.43
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<h2 id="Closing-remarks">Closing remarks<a class="anchor-link" href="#Closing-remarks">¶</a></h2><p>We now see why restarts can provide useful acceleration for Las Vegas algorithms. Hopefully, the generalization to other restartable processes is reasonably clear. In the appendix of this article, there is a lot more discussion of interesting properties and details.</p>
<p>This notebook is available for download <a href="https://github.com/timvieira/blog/blob/master/content/Restart-acceleration.ipynb">here</a>.</p>
<p>I am very happy to answer questions! There are many ways to do that: comment at the end of this document, Tweet at <code>@xtimv</code>, or email <code>tim.f.vieira@gmail.com</code>.</p>
<p>If you found this article interesting please consider sharing it on social media. If you found this article useful please cite it</p>
<div class="highlight"><pre><span></span><span class="nc">@software</span><span class="p">{</span><span class="nl">vieira-restart-acceleration</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">author</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{Tim Vieira}</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">title</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{The restart acceleration trick: A cure for the heavy tail of wasted time}</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">year</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{2019}</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="na">url</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">{https://github.com/timvieira/blog/blob/master/content/Restart-acceleration.ipynb}</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
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<h1 id="Appendix">Appendix<a class="anchor-link" href="#Appendix">¶</a></h1><p>In this section, I have worked out a bunch of lower-level details, which might be of interest to many readers.</p>
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<h3 id="Optimizing-$\tau$">Optimizing $\tau$<a class="anchor-link" href="#Optimizing-$\tau$">¶</a></h3><p>In the discrete case, the optimal $\tau$ can be computed approximately by enumeration.</p>
<p>In the continuous case, the optimal $\tau$ can be computed in "the usual way" (i.e., by solving for $\tau$ such that $\frac{\partial v}{\partial \tau}$ is equal to zero; and check the sign of the second derivative). Setting the derivative equal to zero we get (after a bunch of simplification),</p>
$$
\frac{\partial T}{\partial \tau} = 0
\quad\Longleftrightarrow\quad
\frac{1-p(t \le \tau)}{p(\tau)} - \omega = T
$$
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<p>Remarks</p>
<ul>
<li><p>The optimally capped expectation (as well as higher-order moments) are always finite - even when the underlying RTD puts a positive probability on nontermination.</p>
</li>
<li><p>Optimally tuned thresholds do not always decrease the expected runtime (or higher-order moments). They are most useful in cases where the runtime distribution has a positive probability of nontermination or, more generally, the runtime distribution is heavy-tailed.</p>
</li>
<li><p>The objective is generally nonconvex, so the gradient condition is necessary, but not sufficient.</p>
</li>
</ul>
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<p>Below, we check whether the optimality conditions that we derived are consistent with our running example. The test is an "eyeball" test that the optimal speedup is at the point where the $T(\tau)$ curve (blue) and $\frac{1}{h(t)} -\omega = \frac{1-p(t \le \tau)}{p(\tau)} - \omega$ (orange) intersect. The function $h(t)$ is the ($\omega$-shifted) <a href="https://en.wikipedia.org/wiki/Survival_analysis#Hazard_function_and_cumulative_hazard_function">hazard-rate function</a>.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">tmin</span><span class="p">,</span> <span class="n">tmax</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">Ts</span> <span class="o">=</span> <span class="n">T</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">overhead</span><span class="p">)</span>
<span class="n">H</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">d</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="n">d</span><span class="o">.</span><span class="n">sf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="c1"># hazard-rate function</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">Ts</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'T(R)'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">H</span><span class="p">(</span><span class="n">ts</span><span class="p">)</span><span class="o">-</span><span class="n">overhead</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$1/h(R)-\omega$'</span><span class="p">)</span> <span class="c1"># inverse of the hazard-rate function</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">ts</span><span class="p">[</span><span class="n">Ts</span><span class="o">.</span><span class="n">argmin</span><span class="p">()],</span> <span class="p">[</span><span class="n">Ts</span><span class="o">.</span><span class="n">min</span><span class="p">()],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">75</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="n">ts</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">ts</span><span class="o">.</span><span class="n">max</span><span class="p">())</span> <span class="c1">#; pl.ylim(0, m*2)</span>
<span class="n">rstar</span> <span class="o">=</span> <span class="n">ts</span><span class="p">[</span><span class="n">Ts</span><span class="o">.</span><span class="n">argmin</span><span class="p">()]</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'speedup: $\frac</span><span class="si">{E[T]}{E[T*]}</span><span class="s1">$ = </span><span class="si">%g</span><span class="s1">x faster'</span> <span class="o">%</span> <span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">/</span><span class="n">Ts</span><span class="o">.</span><span class="n">min</span><span class="p">()))</span>
<span class="n">pl</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">H</span><span class="p">(</span><span class="n">rstar</span><span class="p">)</span><span class="o">-</span><span class="n">overhead</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">'--'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$T(R^*)$'</span><span class="p">);</span>
<span class="n">pl</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$\mu = T(\infty)$'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">d</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="o">*</span><span class="mi">2</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'R'</span><span class="p">);</span> <span class="c1">#pl.xscale('log'); pl.yscale('log')</span>
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<h3 id="Runtime-distribution-conditioned-on-capping">Runtime distribution conditioned on capping<a class="anchor-link" href="#Runtime-distribution-conditioned-on-capping">¶</a></h3><p>To study our restarting trick, it will be useful to have the CDF of the $(\tau, \omega)$-restarted distribution. Later in this document, I work out the generalization of this distribution functions for non-stationary restart policies such as the universal sequence.</p>
<p>The restarted distribution has some interesting periodic structure, which we'll discuss below.</p>
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<h4 id="CDF-$F_{\tau,-\omega}(t)$">CDF $F_{\tau, \omega}(t)$<a class="anchor-link" href="#CDF-$F_{\tau,-\omega}(t)$">¶</a></h4>
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<div class="prompt input_prompt">In [10]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">capped_cdf</span><span class="p">(</span><span class="n">τ</span><span class="p">,</span> <span class="n">ω</span><span class="p">):</span>
<span class="s2">"Analytical CDF conditioned on a capping threshold τ and overhead ω."</span>
<span class="n">decay</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">sf</span><span class="p">(</span><span class="n">τ</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">F</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
<span class="c1"># Careful handling of gaps due to restarting and overhead.</span>
<span class="n">k</span><span class="p">,</span><span class="n">r</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">τ</span> <span class="o">+</span> <span class="n">ω</span><span class="p">)</span> <span class="c1"># number of restarts</span>
<span class="k">if</span> <span class="n">r</span> <span class="o"><=</span> <span class="n">τ</span><span class="p">:</span> <span class="c1"># overhead portion of the restart</span>
<span class="c1"># distribution is scaled (by d.sf(τ)**k) and clipped (r <= τ <= τ + ω)</span>
<span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">decay</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">d</span><span class="o">.</span><span class="n">sf</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">decay</span><span class="o">**</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">F</span>
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<h4 id="PDF-$f_{\tau,-\omega}(t)$">PDF $f_{\tau, \omega}(t)$<a class="anchor-link" href="#PDF-$f_{\tau,-\omega}(t)$">¶</a></h4>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">capped_pdf</span><span class="p">(</span><span class="n">τ</span><span class="p">,</span> <span class="n">ω</span><span class="p">):</span>
<span class="s2">"Analytical PDF conditioned on a capping threshold τ and overhead ω."</span>
<span class="n">decay</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">sf</span><span class="p">(</span><span class="n">τ</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
<span class="n">k</span><span class="p">,</span><span class="n">r</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">τ</span> <span class="o">+</span> <span class="n">ω</span><span class="p">)</span>
<span class="k">if</span> <span class="n">r</span> <span class="o"><=</span> <span class="n">τ</span><span class="p">:</span>
<span class="k">return</span> <span class="n">decay</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">d</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">f</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">d</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="mf">.9</span><span class="p">),</span> <span class="mi">10000</span><span class="p">)</span>
<span class="c1"># there should be gaps in the PDF at regular intervals</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">ts</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">/</span> <span class="p">(</span><span class="n">overhead</span> <span class="o">+</span> <span class="n">rstar</span><span class="p">))</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="n">tt</span> <span class="o">=</span> <span class="n">i</span> <span class="o">*</span> <span class="p">(</span><span class="n">overhead</span> <span class="o">+</span> <span class="n">rstar</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">axvspan</span><span class="p">(</span><span class="n">tt</span><span class="o">-</span><span class="n">overhead</span><span class="p">,</span> <span class="n">tt</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.1</span><span class="p">)</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">capped_pdf</span><span class="p">(</span><span class="n">rstar</span><span class="p">,</span> <span class="n">overhead</span><span class="p">)</span>
<span class="n">ps</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">P</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">])</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">ps</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$f_\tau(t)$'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">);</span> <span class="c1">#pl.yscale('log')</span>
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<h3 id="Periodicity-in-the-capped-PDF">Periodicity in the capped PDF<a class="anchor-link" href="#Periodicity-in-the-capped-PDF">¶</a></h3><p>The truncated PDF has an interesting repeating structure. The original distribution's $\tau$-truncated PDF repeats itself every $\tau+\omega$ units of time. However, when it does repeat itself, shrinks by a factor of $F(\tau)$ each time.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">truncated_pdf</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
<span class="k">return</span> <span class="p">(</span><span class="n">a</span> <span class="o"><=</span> <span class="n">t</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">t</span> <span class="o"><=</span> <span class="n">b</span><span class="p">)</span> <span class="o">*</span> <span class="n">d</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
<span class="k">return</span> <span class="n">f</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">period</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">extra</span> <span class="o">=</span> <span class="mf">0.25</span>
<span class="n">λ</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">rstar</span><span class="p">))</span>
<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">((</span><span class="n">period</span> <span class="o">+</span> <span class="mi">0</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">rstar</span> <span class="o">+</span> <span class="n">overhead</span><span class="p">)</span> <span class="o">-</span> <span class="n">extra</span><span class="p">,</span>
<span class="p">(</span><span class="n">period</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">rstar</span> <span class="o">+</span> <span class="n">overhead</span><span class="p">)</span> <span class="o">+</span> <span class="n">extra</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="p">[</span><span class="n">P</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$f_\tau(t)$'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'b'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="c1"># Show periodic self-similarity </span>
<span class="c1"># the next period's in PDF is rescaled (by inverse decay) and shifted one period (rstar + overhead)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="o">/</span><span class="n">λ</span> <span class="o">*</span> <span class="n">P</span><span class="p">(</span><span class="n">t</span> <span class="o">+</span> <span class="p">(</span><span class="n">rstar</span> <span class="o">+</span> <span class="n">overhead</span><span class="p">))</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">],</span>
<span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.5</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">'--'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="c1"># Show the relationship to d's (0,τ*)-truncated PDF.</span>
<span class="n">pl</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">λ</span><span class="p">)</span> <span class="o">*</span> <span class="n">λ</span><span class="o">**</span><span class="n">period</span> <span class="o">*</span> <span class="n">truncated_pdf</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">rstar</span><span class="p">)(</span><span class="n">ts</span> <span class="o">-</span> <span class="p">(</span><span class="n">period</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">rstar</span> <span class="o">+</span> <span class="n">overhead</span><span class="p">)),</span>
<span class="n">color</span><span class="o">=</span><span class="s1">'g'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">0</span><span class="p">);</span>
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<h3 id="Capping-eliminates-heavy-tails">Capping eliminates heavy tails<a class="anchor-link" href="#Capping-eliminates-heavy-tails">¶</a></h3>
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<p>log-log survival plot: This plot shows us if we have heavy tails: a heavy-tailed distribution will have a linear slope, and a light-tailed distribution will go to zero more quickly than linear.</p>
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<div class="prompt input_prompt">In [15]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">tmin</span><span class="p">,</span> <span class="n">d</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="mf">.99</span><span class="p">),</span> <span class="mi">10000</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">ts</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'F(t)'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="o">-</span><span class="n">capped_cdf</span><span class="p">(</span><span class="n">rstar</span><span class="p">,</span> <span class="n">overhead</span><span class="p">)(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$F_\tau(t)$'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s1">'log'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">xscale</span><span class="p">(</span><span class="s1">'log'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'t'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'Survival'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">);</span>
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<h2 id="Tests-for-capped-CDF-and-PDF">Tests for capped CDF and PDF<a class="anchor-link" href="#Tests-for-capped-CDF-and-PDF">¶</a></h2><p>Get a large sample from the restart distribution.</p>
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<div class="prompt input_prompt">In [16]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">simulate</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">seeds</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">_simulate</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">seed</span><span class="p">):</span>
<span class="s2">"simulate capped-restart distribution"</span>
<span class="k">with</span> <span class="n">restore_random_state</span><span class="p">(</span><span class="n">seed</span><span class="p">):</span>
<span class="n">total</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">r</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">rvs</span><span class="p">()</span>
<span class="n">total</span> <span class="o">+=</span> <span class="nb">min</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
<span class="k">if</span> <span class="n">r</span> <span class="o"><=</span> <span class="n">R</span><span class="p">:</span>
<span class="k">return</span> <span class="n">total</span>
<span class="n">total</span> <span class="o">+=</span> <span class="n">overhead</span>
<span class="k">return</span> <span class="p">[</span><span class="n">_simulate</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">seed</span><span class="p">)</span> <span class="k">for</span> <span class="n">seed</span> <span class="ow">in</span> <span class="n">seeds</span><span class="p">]</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">M</span> <span class="o">=</span> <span class="mi">40000</span>
<span class="n">seeds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">**</span><span class="mi">32</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">M</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">simulate</span><span class="p">(</span><span class="n">rstar</span><span class="p">,</span> <span class="n">seeds</span><span class="p">)</span>
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<p>Compare the analytical CDF to the empirical CDF of the big sample.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">cdf</span>
<span class="n">e</span> <span class="o">=</span> <span class="n">cdf</span><span class="p">(</span><span class="n">S</span><span class="p">)</span> <span class="c1"># empirical CDF of the optimal threshold.</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">C</span> <span class="o">=</span> <span class="n">capped_cdf</span><span class="p">(</span><span class="n">rstar</span><span class="p">,</span> <span class="n">overhead</span><span class="p">)</span>
<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">tmin</span><span class="p">,</span> <span class="n">tmax</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span> <span class="c1"># estimate gets unreliable out in the tail</span>
<span class="n">Cs</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="o">-</span><span class="n">C</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">]</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">Cs</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$F_\tau(t)$'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'b'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">e</span><span class="p">(</span><span class="n">ts</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s1">'$\widehat</span><span class="si">{F}</span><span class="s1">_\tau(t)$'</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s1">'log'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">xscale</span><span class="p">(</span><span class="s1">'log'</span><span class="p">);</span>
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<p>Compare the analytical PDF to a histogram fit to the big sample.</p>
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<div class="prompt input_prompt">In [20]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">P</span> <span class="o">=</span> <span class="n">capped_pdf</span><span class="p">(</span><span class="n">rstar</span><span class="p">,</span> <span class="n">overhead</span><span class="p">)</span>
<span class="n">ps</span> <span class="o">=</span> <span class="p">[</span><span class="n">P</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">]</span>
<span class="c1"># Show the histogram</span>
<span class="n">pl</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="n">ts</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">ts</span><span class="o">.</span><span class="n">max</span><span class="p">())</span>
<span class="c1"># The PDF should equal the gradient of the CDF.</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">ps</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">);</span>
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<h3 id="Tests-for-truncated-mean">Tests for truncated mean<a class="anchor-link" href="#Tests-for-truncated-mean">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># test that the right-truncated mean estimate matches rejection sampling</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="c1"># a more efficient estimator would generate truncated samples so all samples are 'accepted' </span>
<span class="k">def</span> <span class="nf">simulate_tm</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
<span class="k">return</span> <span class="n">mean_confidence_interval</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">S</span> <span class="o"><=</span> <span class="n">t</span><span class="p">])</span>
<span class="n">mi</span><span class="p">,</span><span class="n">lb</span><span class="p">,</span><span class="n">ub</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">simulate_tm</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">])</span>
<span class="n">pl</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">lb</span><span class="p">,</span> <span class="n">ub</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">mi</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'b'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">conditional_mean</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">ts</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">);</span>
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<h3 id="Alternative-expressions-for-expected-capped-runtime-$T$">Alternative expressions for expected capped runtime $T$<a class="anchor-link" href="#Alternative-expressions-for-expected-capped-runtime-$T$">¶</a></h3><p>I have used the expression below for expected capped runtime.</p>
$$
T(\tau,\omega) = \mathbb{E}_p[t |t \le \tau] + \frac{1-p(t \le \tau)}{p(t \le \tau)} \cdot (\omega + \tau).
$$<p>However, a number of papers on the subject make use of the following equation,</p>
$$
T(\tau, \omega)
= \frac{\tau - \int_{0}^\tau F(t) \mathrm{d} t}{ F(\tau) } + \omega \frac{1-F(\tau)}{F(\tau)}
$$<p>We can also express $T$ in terms of the survival function $S(t) \overset{\text{def}}{=} (1-F(t))$</p>
$$
T(\tau, \omega)
= \frac{\int_{0}^\tau S(t) \mathrm{d} t}{ 1 - S(\tau) } + \omega \frac{S(\tau)}{1-S(\tau)}
$$
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<div class=" highlight hl-ipython3"><pre><span></span><span class="nd">@np</span><span class="o">.</span><span class="n">vectorize</span>
<span class="k">def</span> <span class="nf">L</span><span class="p">(</span><span class="n">τ</span><span class="p">,</span> <span class="n">ω</span><span class="p">):</span>
<span class="s2">"Equivalent expression for expected runtime given the threshold `b`."</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">τ</span><span class="p">)</span>
<span class="k">return</span> <span class="p">(</span><span class="n">τ</span> <span class="o">-</span> <span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">d</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">τ</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span> <span class="o">/</span> <span class="n">p</span> <span class="o">+</span> <span class="n">ω</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">)</span> <span class="o">/</span> <span class="n">p</span>
<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="mf">.05</span><span class="p">),</span> <span class="n">d</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="mf">.9</span><span class="p">),</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">Ts</span> <span class="o">=</span> <span class="n">T</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">overhead</span><span class="p">)</span>
<span class="n">Ls</span> <span class="o">=</span> <span class="n">L</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">overhead</span><span class="p">)</span>
<span class="c1">#pl.plot(ts, Ls, label=r'$L(R)$', c='r', alpha=0.5)</span>
<span class="c1">#pl.plot(ts, Ts, label=r'$T(R)$', c='b', alpha=0.5)</span>
<span class="n">rerr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">Ts</span> <span class="o">-</span> <span class="n">Ls</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">Ts</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">rerr</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o"><=</span> <span class="mf">0.01</span><span class="p">,</span> <span class="n">rerr</span>
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<h2 id="PDF-and-CDF-of-non-uniform-policies">PDF and CDF of non-uniform policies<a class="anchor-link" href="#PDF-and-CDF-of-non-uniform-policies">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">M</span> <span class="o">=</span> <span class="mi">20000</span>
<span class="n">U_args</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">seeds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">**</span><span class="mi">32</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">M</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span> <span class="c1"># use common random numbers</span>
<span class="n">U</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">_simulate</span><span class="p">(</span><span class="n">universal</span><span class="p">(</span><span class="o">*</span><span class="n">U_args</span><span class="p">)</span><span class="o">.</span><span class="fm">__next__</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">iterview</span><span class="p">(</span><span class="n">seeds</span><span class="p">)])</span>
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<pre>100.0% (20000/20000) [================================================] 00:00:02
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">class</span> <span class="nc">CachedPolicy</span><span class="p">:</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">seq</span><span class="p">,</span> <span class="n">overhead</span><span class="p">,</span> <span class="n">tmax</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span> <span class="bp">self</span><span class="o">.</span><span class="n">s</span> <span class="o">=</span> <span class="p">[];</span> <span class="bp">self</span><span class="o">.</span><span class="n">t</span> <span class="o">=</span> <span class="p">[];</span> <span class="bp">self</span><span class="o">.</span><span class="n">decay</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">T</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span> <span class="n">D</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">():</span>
<span class="k">if</span> <span class="n">T</span> <span class="o">></span> <span class="n">tmax</span><span class="p">:</span> <span class="k">break</span>
<span class="bp">self</span><span class="o">.</span><span class="n">s</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">decay</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
<span class="n">D</span> <span class="o">*=</span> <span class="n">d</span><span class="o">.</span><span class="n">sf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">T</span> <span class="o">+=</span> <span class="n">x</span> <span class="o">+</span> <span class="n">overhead</span>
<span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="bp">self</span><span class="o">.</span><span class="n">overhead</span> <span class="o">=</span> <span class="n">overhead</span>
<span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">s</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">invert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o"><=</span> <span class="n">t</span> <span class="o"><</span> <span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">[</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">]:</span> <span class="c1"># could use binary search for efficiency</span>
<span class="k">return</span> <span class="n">k</span><span class="p">,</span> <span class="n">t</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
<span class="k">assert</span> <span class="kc">False</span><span class="p">,</span> <span class="sa">f</span><span class="s1">'failed to invert policy t=</span><span class="si">{</span><span class="n">t</span><span class="si">}</span><span class="s1"> tmax=</span><span class="si">{</span><span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">)</span><span class="si">}</span><span class="s1">'</span>
<span class="k">def</span> <span class="nf">plot</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="s2">"Show the gaps and restart periods"</span>
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">:</span>
<span class="n">pl</span><span class="o">.</span><span class="n">axvspan</span><span class="p">(</span><span class="n">t</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">overhead</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">pdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
<span class="s2">"Analytical PDF conditioned on a capping policy τ and overhead ω."</span>
<span class="n">k</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="k">if</span> <span class="n">r</span> <span class="o"><=</span> <span class="bp">self</span><span class="o">.</span><span class="n">s</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="p">:</span> <span class="c1"># Are we in an overhead gap period?</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">decay</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">*</span> <span class="n">d</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="k">def</span> <span class="nf">cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
<span class="s2">"Analytical CDF conditioned on a capping threshold τ and overhead ω."</span>
<span class="n">k</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="k">if</span> <span class="n">r</span> <span class="o"><=</span> <span class="bp">self</span><span class="o">.</span><span class="n">s</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="p">:</span> <span class="c1"># Are we in an overhead gap period?</span>
<span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">decay</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">*</span> <span class="n">d</span><span class="o">.</span><span class="n">sf</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">decay</span><span class="p">[</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="mf">.01</span><span class="p">),</span> <span class="n">d</span><span class="o">.</span><span class="n">ppf</span><span class="p">(</span><span class="mf">.95</span><span class="p">),</span> <span class="mi">3000</span><span class="p">)</span>
<span class="n">luby</span> <span class="o">=</span> <span class="n">CachedPolicy</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">universal</span><span class="p">(</span><span class="o">*</span><span class="n">U_args</span><span class="p">),</span> <span class="n">overhead</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">ts</span><span class="o">.</span><span class="n">max</span><span class="p">())</span>
<span class="n">luby</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="p">[</span><span class="n">luby</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.5</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">1.25</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">ts</span><span class="o">.</span><span class="n">max</span><span class="p">());</span>
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<div class="prompt input_prompt">In [30]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">Ucdf</span> <span class="o">=</span> <span class="n">cdf</span><span class="p">(</span><span class="n">U</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="n">d</span><span class="o">.</span><span class="n">sf</span><span class="p">(</span><span class="n">ts</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s1">'original'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="o">-</span><span class="n">C</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">'optimized'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="o">-</span><span class="n">Ucdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">'universal empirical'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'b'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.5</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">':'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ts</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="o">-</span><span class="n">luby</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">.5</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">xscale</span><span class="p">(</span><span class="s1">'log'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s1">'log'</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="n">ts</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">ts</span><span class="o">.</span><span class="n">max</span><span class="p">());</span> <span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s1">'best'</span><span class="p">);</span>
</pre></div>
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</script>Faster reservoir sampling by waiting2019-06-11T00:00:00-04:002019-06-11T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2019-06-11:/blog/post/2019/06/11/faster-reservoir-sampling-by-waiting/<p>We are interested in designing an efficient algorithm for sampling from a categorical distribution over <span class="math">\(n\)</span> items with weights <span class="math">\(w_i > 0\)</span>. Define target sampling distribution <span class="math">\(p\)</span> as
</p>
<div class="math">$$
p = \mathrm{Categorical}\left( \frac{1}{W} \cdot \vec{w} \right)
\quad\text{where}\quad W = \sum_j w_j
$$</div>
<p>The following is a very simple and relatively famous algorithm due to <a href="https://www.sciencedirect.com/science/article/pii/S002001900500298X">Efraimidis and Spirakis (2006)</a>. It has several useful properties (e.g., it is a one-pass "streaming" algorithm, separates data from noise, can be easily extended for streaming sampling without replacement). It is also very closely related to the Gumbel-max trick (<a href="http://timvieira.github.io/blog/post/2014/08/01/gumbel-max-trick-and-weighted-reservoir-sampling/">Vieira, 2014</a>).</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">weighted_reservoir_sampling</span><span class="p">(</span><span class="n">stream</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">([</span><span class="n">Exponential</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">stream</span><span class="p">])</span>
</code></pre></div>
<p>Some cosmetic differences from E&S'06: We use exponential random variates and <span class="math">\(\min\)</span> instead of <span class="math">\(\max\)</span>. E&S'06 use a less-elegant and rather-mysterious (IMO) random key <span class="math">\(u_i^{1/w_i}\)</span>.</p>
<p><strong>Why does it work?</strong> The weighted-reservoir sampling algorithm exploits the following well-known properties of exponential random variates:
When <span class="math">\(X_i \sim \mathrm{Exponential}(w_i)\)</span>, <span class="math">\(R = {\mathrm{argmin}}_i X_i\)</span>, and <span class="math">\(T = \min_i X_i\)</span> then
<span class="math">\(R \sim p\)</span> and <span class="math">\(T \sim \mathrm{Exponential}\left( \sum_i w_i \right)\)</span>.</p>
<h2>Fewer random variates by waiting</h2>
<p>One down-side of this one-pass algorithm is that it requires <span class="math">\(\mathcal{O}(n)\)</span> uniform random variates. Contrast that with the usual, two-pass methods for sampling from a categorical distribution, which only need <span class="math">\(\mathcal{O}(1)\)</span> samples. E&S'06 also present a much less well-known algorithm, called the "Exponential jumps" algorithm, which is a one-pass algorithm that only requires <span class="math">\(\mathcal{O}(\log(n))\)</span> random variates (in expectation). That's <em>way</em> fewer random variates and a small price to pay if you are trying to avoid paging-in data from disk a second time.</p>
<p>Here is my take on their algorithm. There is no substantive difference, but I believe my version is more instructive since it makes the connection to exponential variates and truncated generation explicit (i.e., no mysterious random keys).</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">jump</span><span class="p">(</span><span class="n">stream</span><span class="p">):</span>
<span class="s2">"Weighted-reservoir sampling by jumping"</span>
<span class="n">R</span> <span class="o">=</span> <span class="kc">None</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">inf</span>
<span class="n">J</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">stream</span><span class="p">):</span>
<span class="n">J</span> <span class="o">-=</span> <span class="n">w</span>
<span class="k">if</span> <span class="n">J</span> <span class="o"><=</span> <span class="mi">0</span><span class="p">:</span>
<span class="c1"># Sample the key for item i, given that it is smaller than the current threshold</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">Exponential</span><span class="o">.</span><span class="n">sample_truncated</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
<span class="c1"># i enters the reservoir</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">i</span>
<span class="c1"># sample the waiting time (size of the jump)</span>
<span class="n">J</span> <span class="o">=</span> <span class="n">Exponential</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
<span class="k">return</span> <span class="n">R</span>
</code></pre></div>
<p><strong>Why does exponential jumps work?</strong></p>
<p>Let me first write the <code>weighted_reservoir_sampling</code> algorithm to be much more similar to the <code>jump</code> algorithm. For fun, I'm going to refer to it as the <code>walk</code> algorithm.</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">walk</span><span class="p">(</span><span class="n">stream</span><span class="p">):</span>
<span class="s2">"Weighted-reservoir sampling by walking"</span>
<span class="n">R</span> <span class="o">=</span> <span class="kc">None</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">inf</span>
<span class="n">J</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">stream</span><span class="p">):</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">Exponential</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="k">if</span> <span class="n">X</span> <span class="o"><</span> <span class="n">T</span><span class="p">:</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">i</span> <span class="c1"># i enters the reservoir</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">X</span> <span class="c1"># threshold to enter the reservoir</span>
<span class="k">return</span> <span class="n">R</span>
</code></pre></div>
<p><strong>The key idea</strong> of the exponential jumps algorithm is to sample <em>waiting times</em> between new minimum events. In particular, if the algorithm is at step <span class="math">\(i\)</span> the probability that it sees its next minimum at steps <span class="math">\(j \in \{ i+1, \ldots \}\)</span> can be reasoned about without needing to <em>actually</em> sample the various <span class="math">\(X_j\)</span> variables.</p>
<p>Rather than going into a full-blown tutorial on waiting times of exponential variates, I will get to the point and show that the <code>jump</code> algorithm simulates the <code>walk</code> algorithm. The key to doing this is showing that the probability of jumping from <span class="math">\(i\)</span> to <span class="math">\(k\)</span> is the same as "walking" from <span class="math">\(i\)</span> to <span class="math">\(k\)</span>. Let <span class="math">\(W_{i,k} = \sum_{j=i}^k w_j\)</span>.</p>
<p>This proof is adapted from the original proof in E&S'06.</p>
<div class="math">$$
\begin{eqnarray}
\mathrm{Pr}\left( \text{walk to } k \mid i,T \right)
&=& \mathrm{Pr}\left( X_k < T \right) \prod_{j=i}^{k-1} \mathrm{Pr}\left( X_j \ge T \right) \\
&=& \left(1 - \exp\left( T w_k \right) \right) \prod_{j=i}^{k-1} \exp\left( T w_j \right) \\
&=& \left(1 - \exp\left( T w_k \right) \right) \exp\left( T \sum_{j=i}^{k-1} w_j \right) \\
&=& \left(1 - \exp\left( T w_k \right) \right) \exp\left( T W_{i,k-1} \right) \\
&=& \exp\left( T W_{i,k-1} \right) - \exp\left( T w_k \right) \exp\left( T W_{i,k-1} \right) \\
&=& \exp\left( T W_{i,k-1} \right) - \exp\left( T W_{i,k} \right) \\
\\
\mathrm{Pr}\left( \text{jump to } k \mid i, T \right)
&=& \mathrm{Pr}\left( W_{i,k-1} < J \le W_{i,k} \right) \\
&=& \mathrm{Pr}\left( W_{i,k-1} < -\frac{\log(U)}{T} \le W_{i,k} \right) \\
&=& \mathrm{Pr}\left( \exp(-T W_{i,k-1}) > U \ge \exp(-T W_{i,k}) \right) \label{foo}\\
&=& \exp(T W_{i,k-1}) - \exp(T W_{i,k} )
\end{eqnarray}
$$</div>
<p>Given that the waiting time correctly matches the walking algorithm, the remaining detail is to check that <span class="math">\(X_k\)</span> is equivalent under the condition that it goes into the reservoir. This conditioning is why the jumping algorithm must generate a <em>truncated</em> random variate: a random variate that is guaranteed to less than the previous minimum. In the <a href="https://cmaddis.github.io/gumbel-machinery">Gumbel-max world</a>, this is used in the top-down generative story.</p>
<h2>Closing thoughts</h2>
<p>Pros:</p>
<ul>
<li>The jump algorithm saves a ton of random variates and gives practical savings
(at least, in my limited experiments).</li>
</ul>
<p>Cons:</p>
<ul>
<li>
<p>The jump algorithm is harder to parallelize or vectorize, but it seems possible.</p>
</li>
<li>
<p>If you aren't in a setting that requires a one-pass algorithm or some other
special properties, you are probably better served by the two-pass algorithms
since they have lower overhead because it doesn't call expensive functions
like <span class="math">\(\log\)</span> and it uses a single random variate per sample.</p>
</li>
</ul>
<p>Further reading:</p>
<ul>
<li>
<p>I have several posts on the topic of fast sampling algorithms
(<a href="http://timvieira.github.io/blog/post/2016/11/21/heaps-for-incremental-computation/">1</a>,
<a href="http://timvieira.github.io/blog/post/2016/07/04/fast-sigmoid-sampling/">2</a>,
<a href="http://timvieira.github.io/blog/post/2014/08/01/gumbel-max-trick-and-weighted-reservoir-sampling/">3</a>).</p>
</li>
<li>
<p>Jake VanderPlas (2018) <a href="http://jakevdp.github.io/blog/2018/09/13/waiting-time-paradox/">The Waiting Time Paradox, or, Why Is My Bus Always Late?</a>.</p>
</li>
</ul>
<h2>Interactive Notebook</h2>
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</script>The likelihood-ratio gradient2019-04-20T00:00:00-04:002019-04-20T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2019-04-20:/blog/post/2019/04/20/the-likelihood-ratio-gradient/<p><strong>Setup</strong>: We're trying to optimize a function of the form</p>
<div class="math">$$
J(\theta) = \underset{p_\theta}{\mathbb{E}} \left[ r(x) \right] = \sum_{x \in \mathcal{X}} p_\theta(x) r(x).
$$</div>
<p>The problem is that we can't just evaluate each <span class="math">\(x \in \mathcal{X}\)</span> because we
don't have complete knowledge of <span class="math">\(p_\theta\)</span>. For example, it is a mix of
factors that are known and under our control via <span class="math">\(\theta\)</span> (policy factors) and
factors that are not known (environment factors).</p>
<p>Combined with stochastic gradient ascent, the likelihood-ratio gradient estimator is an approach for solving such a
problem. It appears in policy gradient methods for reinforcement learning
(e.g.,
<a href="https://papers.nips.cc/paper/1713-policy-gradient-methods-for-reinforcement-learning-with-function-approximation.pdf">Sutton et al. 1999</a>),
black-box optimization (e.g., <a href="https://arxiv.org/abs/1106.4487">Wierstra et al. 2011</a>), and <a href="https://timvieira.github.io/blog/post/2016/12/19/counterfactual-reasoning-and-learning-from-logged-data/">causal reasoning</a>. There are two main ideas in the
trick: (1) the "score function" estimator and (2) the cancelation of
complicating factors.</p>
<h4>Part 1: The score function gradient estimator</h4>
<p>Suppose we can sample <span class="math">\(x^{(j)} \sim p_\theta\)</span>. This opens up the following
(unbiased) Monte Carlo estimators for <span class="math">\(J\)</span> and its gradient,</p>
<div class="math">$$
J(\theta) \approx \frac{1}{m} \sum_{j=1}^m r(x^{(j)})
$$</div>
<div class="math">$$
\nabla_{\!\theta} J(\theta) \approx \frac{1}{m} \sum_{j=1}^m r(x^{(j)}) \nabla_{\!\theta} \log p_{\theta}(x^{(j)}).
$$</div>
<p>The derivation is pretty simple
</p>
<div class="math">$$
\begin{eqnarray*}
\nabla_{\!\theta} \, \underset{p_\theta}{\mathbb{E}}\left[ r(x) \right]
&=& \nabla_{\!\theta} \left[ \sum_x p_{\theta}(x) r(x) \right] \\
&=& \sum_x \nabla_{\!\theta} \left[ p_{\theta}(x) \right] r(x) \\
&=& \sum_x p_{\theta}(x) \frac{\nabla_{\!\theta} \left[ p_{\theta}(x) \right] }{ p_{\theta}(x) } r(x) \\
&=& \underset{p}{\mathbb{E}}\left[ r(x) \nabla_{\!\theta} \log p_\theta(x) \right]
\end{eqnarray*}
$$</div>
<p><br/>
We use the identity <span class="math">\(\nabla f = f\, \nabla \log f\)</span>, assuming <span class="math">\(f > 0\)</span>.</p>
<p>To use this estimator, we only need two things (1) the ability to sample
<span class="math">\(x^{(j)} \sim p_{\theta}\)</span>, (2) the ability to evaluate <span class="math">\(\log
p_{\theta}(x^{(j)})\)</span> and <span class="math">\(r(x^{(j)})\)</span> for each sampled value.</p>
<p>This isn't even the entire method, but we can already use it to do some neat
things. For example, minimum risk training of structured prediction models.
Assuming we can obtain good samples—preferably exact samples, but MCMC
samples might be ok—the likelihood ratio can help us learning even with
complicated blackbox cost functions (sometimes called "nondecomposable loss
functions") like human annotators or impenetrable perl scripts. I had this idea
back in 2012, but never got around to pushing it out. There appear to be some
papers that picked up on this idea, including
<a href="http://www.cl.uni-heidelberg.de/~riezler/publications/papers/ACL2016.pdf">Sokolov et al. (2016)</a>
and <a href="https://arxiv.org/abs/1609.00150">Norouzi et al. (2016)</a> and even a few
papers using it for "black box" variational inference
<a href="https://arxiv.org/abs/1401.0118">(Ranganath et al., 2013)</a>.</p>
<p><em>Remarks:</em></p>
<ul>
<li>
<p><strong>Relaxing discrete actions into stochastic ones</strong>: A common way to handle
discrete decisions is to put a <em>differentiable</em> parametric density (like
<span class="math">\(p_\theta\)</span>) over the space of possible executions (paths <span class="math">\(x\)</span>). (Note: this
shouldn't be surprising—it's already what we do in structured
prediction methods like conditional random fields!) The likelihood-ratio
method can be used to estimate gradients in such settings.</p>
</li>
<li>
<p><strong>Bandit feedback</strong>: This approach naturally handles "bandit feedback"
(partial information about <span class="math">\(r\)</span>): you only see the values of only the
trajectories that you actually sample. In contrast with "full information",
which tells you the reward of all possible trajectories.</p>
</li>
</ul>
<h5>The off-policy estimator</h5>
<p>Let's generalize this estimator to allow off-policy actions
<a href="https://timvieira.github.io/blog/post/2014/12/21/importance-sampling/">importance-weighted estimator</a>. Here
<span class="math">\(q\)</span> is a distribution over the same space as <span class="math">\(p\)</span> with support at least
everywhere <span class="math">\(p\)</span> has support. </p>
<div class="math">$$ \begin{eqnarray*} \nabla_{\!\theta} \,
\underset{p_\theta}{\mathbb{E}}\left[ r(x) \right] &=&
\underset{p}{\mathbb{E}}\left[ r(x) \nabla_{\!\theta} \log p_\theta(x) \right]
\\ &=& \sum_x p_{\theta}(x) r(x) \nabla_{\!\theta} \log p_\theta(x) \\ &=&
\sum_x \frac{q(x)}{q(x)} p_{\theta}(x) r(x) \nabla_{\!\theta} \log p_\theta(x)
\\ &=&
\underset{q}{\mathbb{E}}\left[ \frac{p_{\theta}(x)}{q(x)} r(x) \nabla_{\!\theta} \log p_\theta(x) \right]
\\ &\approx& \frac{1}{n} \sum_{i=1}^n \frac{p_{\theta}(x^{(i)})}{q(x^{(i)})} r(x^{(i)}) \nabla_{\!\theta} \log p_\theta(x^{(i)})\quad \text{ where } x^{(i)} \sim q
\end{eqnarray*} $$</div>
<p>Note that we recover the original estimator when <span class="math">\(q=p\)</span>.</p>
<h4>Part 2: The convenient cancelation of complicating components</h4>
<p>The real power of the <em>likelihood-ratio</em> part of this method comes when you have
the ability to sample <span class="math">\(x\)</span>, but <em>not</em> the ability to compute the probability of
<em>all</em> factors of the joint probability of <span class="math">\(x\)</span> (i.e., you can't compute the
<em>complete</em> score <span class="math">\(p_{\theta}(x)\)</span>). In other words, some components of the joint
probability's <em>generative process</em> might pass through factors which are <em>only
accessible through sampling</em>, e.g., because they require performing <em>actual
experiments</em> in the real world or a complex simulation! The factors that we can
only sample from are what make this a true stochastic optimization problem.</p>
<p>Let's be a little more concrete by looking at a classic example from
reinforcement learning: the Markov decision process (MDP). In this context, the
random variable <span class="math">\(x\)</span> is an alternating sequence of states and actions, <span class="math">\(x =
\langle s_0, a_0, s_1, a_1, \ldots a_{T-1}, s_T \rangle\)</span> and the generative
process consists of an unknown transition function <span class="math">\(p(s_{t+1}|s_t,a_t)\)</span> that is
only accessible through sampling and a policy <span class="math">\(p_{\theta}(a_t|s_t)\)</span> which we in
control of. So the probability of an entire sequence in an MDP <span class="math">\(p_{\theta}(x)\)</span>
is <span class="math">\(p(s_0) \prod_{t=0}^T p(s_{t+1}|s_t,a_t) \pi_\theta(a_t|s_t)\)</span>. The
likelihood-ratio method can be used to derive several "policy gradient" methods,
which compute unbiased gradient estimates with no knowledge of the transition
distribution.</p>
<blockquote>
<p>The beauty of the likelihood ratio is the cancellation of unknown terms.</p>
</blockquote>
<p>Aside: This fortunate cancellation occurs in many other contexts, e.g. the
Metropolis-Hastings accept-reject criteria.</p>
<p>To make this explicit, let's consider the importance weight, <span class="math">\(p/q\)</span>.
</p>
<div class="math">\begin{eqnarray}
\frac{p_\theta(x)}{q(x)}
= \frac{ {\color{red}{ p(s_0) }} \prod_{t=0}^T {\color{red}{ p(s_{t+1}|s_t,a_t) }} \pi_\theta(a_t|s_t) }
{ {\color{red}{ p(s_0) }} \prod_{t=0}^T {\color{red}{ p(s_{t+1}|s_t,a_t) }} q(a_t|s_t) }
= \frac{\prod_{t=0}^T \pi_\theta(a_t|s_t)}
{\prod_{t=0}^T q(a_t|s_t)}
\end{eqnarray}</div>
<p><br/>
Common terms cancel! This implies that we don't need to compute them.</p>
<p>Those component cancel in <span class="math">\(\nabla_{\!\theta} \log p_{\theta}(x)\)</span> because terms
that do not depend on <span class="math">\(\theta\)</span> also disappear. Leaving you with just a sum of
log-gradient terms that you <em>do</em> know because they are part of the model you're
tuning.</p>
<div class="math">$$
\begin{eqnarray*}
\nabla \log p(x)
&=& \nabla \log \left( p(s_0) \prod_{t=0}^T p(s_{t+1}|s_t,a_t) \pi_\theta(a_t|s_t) \right) \\
&=& \nabla \left(\log p(s_0) + \sum_{t=0}^T \log p(s_{t+1}|s_t,a_t)
+ \log \pi_\theta(a_t|s_t) \right) \\
&=& \sum_{t=0}^T \nabla \log \pi_\theta(a_t|s_t)
\end{eqnarray*}
$$</div>
<h4>The baseline trick</h4>
<p>These estimators should always be used in conjunction with a baseline function
or more generally a control variate. There are many options for deriving control
variates, which will depend on the specific structure of <span class="math">\(x\)</span>. For example, in
the MDP case, we can use any function that depends on <span class="math">\(s_t\)</span>.</p>
<p>However, even without special structure, we can an always should use (at a
minimum) a constant baseline,
</p>
<div class="math">$$
\mathbb{E}_{x \sim q} \left[
\frac{p_{\theta}(x)}{q(x)}
r(x)
\nabla_{\!\theta} \log p_\theta(x)
\right]
=
\mathbb{E}_{x \sim q} \left[
\frac{p_{\theta}(x)}{q(x)}
(r(x) - {\color{red}{b}})
\nabla_{\!\theta} \log p_\theta(x)
\right]
\text{for all } {\color{red}{b} \in \mathbb{R}}
$$</div>
<p>
The minimum variance choice for b is
</p>
<div class="math">$$
b = \frac{\sum_k \mathrm{Cov}(r, \nabla_{\theta_k} \log p) }{\sum_k \mathrm{Var}(\nabla_{\theta_k} \log p) }
$$</div>
<p>
which we can compute with sampling-based estimators of the quantities.</p>
<p>Some folks use an estimate of <span class="math">\(J\)</span>, which is better than nothing.</p>
<h4>Important points</h4>
<ul>
<li>
<p>Always use a baseline.</p>
</li>
<li>
<p>This gradient estimate is "zero order" it is essentially probing the function
in <span class="math">\(x\)</span> space, which might be higher dimensional than <span class="math">\(\theta\)</span>. As a result,
you might be better off with gradient estimators that are based on perturbing
<span class="math">\(\theta\)</span> directly, e.g., zeroth-order methods (sometimes called <em>direct
search</em> or <em>gradient-free</em> optimization methods) like Nelder-Mead simplex,
FDSA, SPSA, and CMA-ES.</p>
</li>
<li>
<p>Often there is almost no signal. Consider the example of trying to solve a
maze by randomly running around in it. In this case, it's very unlikely that
a random path will lead to a positive outcome. Therefore, the gradient
really is essentially zero. So even with access to the <em>true</em> gradient (i.e.,
no variance), optimization would have a lot of trouble finding a good
optimum. Add to that some variance and you have useless on top of noisy.</p>
</li>
<li>
<p>Although the likelihood-ratio gives us an unbiased estimate of the gradient,
don't be fooled. The particular gradient estimate used in the
likelihood-ratio method has an impractical signal-to-noise ratio, which makes
it very hard use in optimization. There are countless papers on tricks to
reduce the variance of the estimator.</p>
</li>
<li>
<p>You can improve your data efficiency and algorithm stability using off-line
optimization (with your favorite deterministic optimization algorithm). I
have written a long article about offline optimization
<a href="https://timvieira.github.io/blog/post/2016/12/19/counterfactual-reasoning-and-learning-from-logged-data/">here</a>.</p>
</li>
</ul>
<h2>Summary</h2>
<p>There is still a lot to say about likelihood-ratio methods. I didn't talk about
control variates or "baseline" functions, which are very important to making
things work. I'll try to post my notes on those ideas soon.</p>
<p><strong>Take home messages</strong>:</p>
<ul>
<li>
<p>If you can evaluate it, then you can take the gradient of it (assuming it
exists). This even holds if the evaluation is based on Monte Carlo.</p>
</li>
<li>
<p>The likelihood-ratio shows up all over the place, not just RL. It shows up in
<a href="https://timvieira.github.io/blog/post/2016/12/19/counterfactual-reasoning-and-learning-from-logged-data/">causal reasoning</a>
more generally.</p>
</li>
<li>
<p>We described a general way to learn from watching someone else act in a world
we don't understand (i.e., off-policy learning with no knowledge of the
environment just samples!). The only catch is that in order for us to learn
from them we need them to do a little bit of "exploration" (i.e., be a
stochastic policy that has support everywhere we do) and tell us their action
probabilities (so that we can important weight against our policy).</p>
</li>
</ul>
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</script>Steepest ascent2019-04-19T00:00:00-04:002019-04-19T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2019-04-19:/blog/post/2019/04/19/steepest-ascent/<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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<p><strong>Abstract:</strong> In my last post, I talked about <a href="http://timvieira.github.io/blog/post/2018/03/16/black-box-optimization/">black-box optimization</a> where I discussed the idea of "ascent directions" in optimization. In this post, I'm going to discuss what it means to be the <em>steepest</em> ascent direction and what it means to be a "steepest-ascent direction," formally.</p>
<p>Steepest ascent is a nice unifying framework for understanding different optimization algorithms. Almost every optimization algorithm is performing steepest ascent in what way or another—the question is in what <em>space</em>?</p>
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<p><strong>About the format of this post:</strong> In addition to deriving things mathematically, I will also give Python code alongside it. The idea is that the code will directly follow the math. I often <em>simulate</em> math in order to double check my work and avoid silly mistakes, which is super important when working solo on new stuff. Even better and more important: this approach makes math <em>interactive</em>—enabling me to experiment, build intuition, and keep things grounded in actual examples. It also forces me to think about what I'm doing at many levels:</p>
<ol>
<li>formal (math)</li>
<li>intuitive (pictures and explanations)</li>
<li>procedural (code)</li>
<li>grounded (examples and applications)</li>
</ol>
<p>Having many levels at my disposal lets me do "<a href="https://en.wikipedia.org/wiki/Co-training">co-training</a>" to developing my understanding. Sometimes, I even view math as something that needs to be "empirically verified," which is kind of ridiculous, but I think the mindset isn't terrible: always be skeptical. Just because something is nicely typeset, doesn't make it correct.</p>
<p>One of the worst things when trying to learn or experiment with new things (e.g., do research) is a slow turn-around to simply "try" something out. Therefore, I really love tools that facilitate rapid prototyping (e.g., black-box optimizers, automatic & numerical differentiation, and visualization tools).</p>
<p>Put your understanding to the test! Have you worked out all the math? implemented it? Ok, now try writing down a simple illustrative example (think: the idea is "software" in need of testing) that shows the method works as advertised. Now, try to break it.</p>
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<h2 id="What-is-steepest-ascent?">What is steepest ascent?<a class="anchor-link" href="#What-is-steepest-ascent?">¶</a></h2>
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<p><strong>Setup (a generic optimization problem):</strong> We want to maximize a multivariate function $f$ over some space $\mathcal{X}$.</p>
$$
x^* = \underset{x \in \mathcal{X}}{\textrm{argmax }} f(x)
$$
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<p>Most algorithms for approaching this type of problem are iterative, "hill climbing" algorithms, which use information about how the function behaves near the current point to form a <em>search direction</em>. A classic example is, of course, ordinary gradient ascent whose search direction is simply the gradient.</p>
<p>You may have learned in calculus that "the gradient is the direction of steepest ascent." While this is true, it is only true under the assumption that $\mathcal{X}$ is a Euclidean space, i.e., a space where it makes sense to measure the distance between two points with the Euclidean distance. Clearly, not all spaces even type check as Euclidean (e.g., discrete spaces), and in some cases, Euclidean distances ignore important structure and constraints (e.g., probability distributions are positive and integrate to unity).</p>
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<p><strong>Search direction:</strong> We want our algorithms to search in directions, which will result in improvements to the function value. Clearly, the <em>best</em> search direction is just $(x^* - x)$: a simple step of size $1$ lands us at the optimum! Of course, this doesn't help us actually find $x^*$! So we want a more "modest," computationally friendly notion of a search direction. Therefore, we will narrow our attention to local search directions; in particular, steepest-descent directions.</p>
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<p><strong>The steepest direction:</strong> The word "steep" is talking about a slope:</p>
$$
\text{steepness} = \frac{\text{change in }f}{\text{change in }x}
$$<p>The change in $f$ is straightforward to measure because it's a scalar. The change in $x$ is more complicated because there are many ways to compare $x$s—they might be vectors, they may not even be real-valued objects!</p>
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<p><strong>What does it mean to <em>change</em> $x$:</strong> There are countless ways to "change" $x$. The most general case is that of a general operator: $x' = \Delta(x)$, where $\Delta$ is an arbitrary transform of from $\mathcal{X}$ to $\mathcal{X}$. In this post, we will primarily consider <em>additive</em> changes of the form $x' = x + \Delta.$ However, in many contexts, it makes sense to change $x$ in other ways! For example, multiplicative changes $x' = x \cdot \Delta$, or even discrete search moves (e.g., combinatorial problems).</p>
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<p><strong>Measuring the change:</strong> We need a way to measure the size of the change to $x$. Let's assume a "distance function" $d(x, \Delta(x)) \ge 0$. There are some assumptions about what makes a valid distance function, which I won't cover here. To avoid notational clutter, I will write $\rho(\Delta)$ instead of $d(x, \Delta(x))$. Note that $\rho$ has an implicit dependence on $x$, but since that dependence does not affect our discussion, I will leave it implicit.</p>
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<p><strong>Steepest-ascent problem:</strong> The steepest-ascent direction is the solution to the following optimization problem, which a nice generalization of the definition of the derivatives that (1) considers a more general family of changes than additive and (2) a holistic measurement for the change in x,</p>
$$
\Delta^*
= \underset{\Delta}{\textrm{argmax }} \frac{ f(\Delta(x)) - f(x) }{ \rho(\Delta) }.
$$<p>Unfortunately, this optimization problem is "nasty" because it contains a ratio that includes a change with $\rho(\Delta)=0$. Therefore, we need to consider a limit analysis to make sense of the division by zero. In particular, we will consider the limit of the following approximation via a constraint $\rho(\Delta) \le \varepsilon$ with $\varepsilon > 0$,</p>
$$
\Delta^*(\varepsilon) = \underset{\rho(\Delta) \le \varepsilon}{\textrm{argmax }} f(\Delta(x)).
$$<p>The when limit $\varepsilon \rightarrow 0^+$ exists, we recover a solution to the original problem,</p>
$$
\lim_{\varepsilon \rightarrow 0^+} \Delta^*(\varepsilon) = \Delta^*.
$$<p>Note that "continuity of argmax," which we need for the limit to exist, is a very technical topic with lots of great theory that we can geek out about at some other time :P</p>
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<p><strong>Generic steepest-ascent algorithm:</strong> We now have a generic steepest-ascent optimization algorithm:</p>
<ol>
<li>Start with a guess $x_0$ and set $t = 0$.</li>
<li>Pick $\varepsilon_t$</li>
<li>Solving the steepest descent problem to get $\Delta_t$ conditioned the current iterate $x_t$ and choice $\varepsilon_t$.</li>
<li>Apply the transform to get the next iterate, $x_{t+1} \leftarrow \textrm{stepsize}( \Delta_t(x_t) )$</li>
<li>Set $t \leftarrow t + 1$</li>
<li>Repeat steps 2-6.</li>
</ol>
<p>Admittedly, this $\textrm{stepsize}$ function is totally mysterious and will depend heavily on all of our design decisions (the changes we've chosen to search for $\Delta$, the value of $\varepsilon$, properties of the space $\mathcal{X}$, and the objective function $f$). Lots of the work needed to ensure convergence and other properties of the algorithm will go into carefully designing this function. For the purposes of this article, assume that there is a mechanism for "abstract line search" that will just make these decisions optimally. Magical line search doesn't guarantee that our algorithm won't get stuck at poor choices of $\mathcal{X}$ (e.g., poor local optima) or even that it won't sit around and oscillate.</p>
<p><strong>Remarks:</strong></p>
<ul>
<li>Why are we looking at the rate of change instead of just change in $f$? <em>Rate</em> of change has a nice analogy to rolling down a hill.</li>
<li>Up to this point, I have made no assumption about the continuity of $f$ or $\mathcal{X}$.</li>
<li>What happens if our family of changes does not maintain feasibility, i.e., $\Delta(x) \notin \mathcal{X}$?</li>
<li>This algorithm is almost too abstract to be useful. It's interesting to see how far you can take an abstract idea (in our case, steepest ascent) before you start needing to make assumptions (e.g., step size).</li>
</ul>
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<h2 id="Steepest-ascent-with-additive-changes">Steepest ascent with additive changes<a class="anchor-link" href="#Steepest-ascent-with-additive-changes">¶</a></h2>
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<p>Up until this point, we have been very abstract and non-committal in developing the steepest-ascent framework. In the section, we will make a few assumptions (below), which will allow us to go a little deeper in studying the steepest-ascent framework.</p>
<ul>
<li>$f$ is a continuous, real-valued function over $\mathcal{X} = \mathbb{R}^n$.</li>
<li>Changes are from an additive parametric family, $\Delta^{\text{additive}}_d(x) = x + d$ where the parameter $d$ is also in $\mathbb{R}^n$.</li>
<li>We will take the limit of the steepest-ascent problem as $\varepsilon \rightarrow 0^+$.</li>
</ul>
<p>These assumptions (plus $\rho$ as the Euclidean norm) are, by far, the most common assumptions made in unconstrained continuous optimization. We will study a few choices for $\rho$, which have interesting interpretations and ramifications for steepest-ascent algorithms.</p>
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<h3 id="Numerical-experiments">Numerical experiments<a class="anchor-link" href="#Numerical-experiments">¶</a></h3>
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<p>Below is a simple implementation of a <em>numerical</em> steepest-descent search algorithm.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="kn">import</span> <span class="n">minimize</span>
<span class="k">def</span> <span class="nf">steepest</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Delta</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">,</span> <span class="n">visualize</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Find the direction of steepest ascent for the function `f`, where the direction</span>
<span class="sd"> is `eps` far away under norm `p` (which implicitly measures the distance from </span>
<span class="sd"> the current iterate `x`).</span>
<span class="sd"> """</span>
<span class="n">opt</span> <span class="o">=</span> <span class="n">minimize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="o">-</span><span class="n">f</span><span class="p">(</span><span class="n">Delta</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">d</span><span class="p">)),</span> <span class="n">x0</span> <span class="o">=</span> <span class="mi">0</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">options</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">maxiter</span> <span class="o">=</span> <span class="mi">10000</span><span class="p">),</span>
<span class="n">constraints</span> <span class="o">=</span> <span class="p">[{</span><span class="s1">'fun'</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">eps</span> <span class="o">-</span> <span class="n">p</span><span class="p">(</span><span class="n">d</span><span class="p">),</span> <span class="s1">'type'</span><span class="p">:</span> <span class="s1">'ineq'</span><span class="p">}])</span>
<span class="k">assert</span> <span class="n">opt</span><span class="o">.</span><span class="n">success</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">message</span>
<span class="k">if</span> <span class="n">visualize</span><span class="p">:</span> <span class="n">plot_steepest</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Delta</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
<span class="k">return</span> <span class="n">opt</span><span class="o">.</span><span class="n">x</span> <span class="c1"># output will have unit-length vector under p</span>
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<p>We can visualize our optimization problem in two dimensions.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">plot_steepest</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Delta</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">opt</span><span class="p">,</span> <span class="n">eps</span><span class="p">):</span>
<span class="n">z</span> <span class="o">=</span> <span class="mf">1.75</span><span class="o">*</span><span class="nb">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">opt</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span> <span class="n">eps</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">z</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mi">100</span><span class="p">]</span>
<span class="n">contour_plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">f</span><span class="p">(</span><span class="n">Delta</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">d</span><span class="p">)),</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
<span class="n">contour_plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="nb">float</span><span class="p">(</span><span class="n">p</span><span class="p">(</span><span class="n">d</span><span class="p">)</span> <span class="o"><=</span> <span class="n">eps</span><span class="p">),</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'binary_r'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="n">opt</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[</span><span class="n">opt</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">);</span>
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<p>Let's test it out on a simple objective function. Feel free to download the notebook and try your own!</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span><span class="o">,</span> <span class="nn">pylab</span> <span class="k">as</span> <span class="nn">pl</span>
<span class="kn">import</span> <span class="nn">numdifftools</span> <span class="k">as</span> <span class="nn">nd</span> <span class="c1"># use numerical derivatives, cuz they are really easy to work with.</span>
<span class="n">D</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">D</span><span class="p">,</span><span class="n">D</span><span class="p">))</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="n">D</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">A</span><span class="o">=</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="n">B</span><span class="p">):</span> <span class="k">return</span> <span class="n">B</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span> <span class="c1"># some aribitrary function with input dimension D</span>
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<p>Below we compare the numerical solution to the steepest-ascent problem (under the Euclidean distance) to the gradient and see that they are equivalent (no surprise here).</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">compare</span><span class="p">,</span> <span class="n">normalize</span>
<span class="kn">from</span> <span class="nn">arsenal.viz</span> <span class="kn">import</span> <span class="n">contour_plot</span>
<span class="kn">from</span> <span class="nn">scipy.linalg</span> <span class="kn">import</span> <span class="n">norm</span>
<span class="n">eps</span> <span class="o">=</span> <span class="mf">1e-5</span>
<span class="n">x0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="n">D</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">norm</span>
<span class="n">Delta</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">d</span><span class="p">:</span> <span class="n">x</span><span class="o">+</span><span class="n">d</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">steepest</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Delta</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">visualize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">g0</span> <span class="o">=</span> <span class="n">nd</span><span class="o">.</span><span class="n">Gradient</span><span class="p">(</span><span class="n">f</span><span class="p">)(</span><span class="n">x0</span><span class="p">)</span>
<span class="n">compare</span><span class="p">(</span><span class="n">d</span> <span class="o">/</span> <span class="n">p</span><span class="p">(</span><span class="n">d</span><span class="p">),</span> <span class="n">g0</span> <span class="o">/</span> <span class="n">p</span><span class="p">(</span><span class="n">g0</span><span class="p">))</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
<span class="c1">#assert np.allclose(d / p(d), g0 / p(g0), rtol=0.05)</span>
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<pre>100.0% (10000/10000) [================================================] 00:00:00
100.0% (10000/10000) [================================================] 00:00:00
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Comparison: n=2
norms: <span class="ansi-green-fg">[1, 1]</span>
zero F1: <span class="ansi-green-fg">1</span>
pearson: <span class="ansi-green-fg">1</span>
spearman: <span class="ansi-green-fg">1</span>
Linf: <span class="ansi-red-fg">0.0021266</span>
same-sign: <span class="ansi-green-fg">100.0% (2/2)</span>
max rel err: <span class="ansi-green-fg">0.00471391</span>
regression: <span class="ansi-yellow-fg">[1.001 -0.002]</span>
got is larger: <span class="ansi-red-fg">100.0% (2/2)</span>
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<h5 id="TL;DR-notes-on-the-code">TL;DR notes on the code<a class="anchor-link" href="#TL;DR-notes-on-the-code">¶</a></h5>
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<li><p>For comparing these directions, I'm using my vector comparison utility <a href="https://github.com/timvieira/arsenal/blob/master/arsenal/math/compare.py">arsenal.math.compare</a>, which gives me a bunch of detail metrics comparing the two vectors. This is sort of overkill for what we're using it for, but it was useful in debugging.</p>
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<li><p>Minor note: I'm using <code>scipy.linalg.norm</code> instead of <code>numpy.linalg.norm</code> because it is more numerically stable (<a href="https://timvieira.github.io/blog/post/2014/11/10/numerically-stable-p-norms/">further reading</a>).</p>
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<h3 id="Let's-look-at-a-slightly-more-interesting-example.">Let's look at a slightly more interesting example.<a class="anchor-link" href="#Let's-look-at-a-slightly-more-interesting-example.">¶</a></h3><p>I've written before about the <a href="https://timvieira.github.io/blog/post/2016/05/27/dimensional-analysis-of-gradient-ascent/">dimensional analysis of gradient descent</a>. In this post, I explain why the step-size parameter in gradient descent is hard to determine a priori because it is not unit free—in fact, its units are pretty complicated. One explanation for why this is the case is that we are using a silly metric on $x$ because the dimensions of $x$ are generally not in the same units, and this should not be compared without some type of common unit conversion!</p>
<p>Ok, let's do that. Suppose that we have a simple rule to map each dimension $x_i$ into a common currency, let's suppose the conversion is $\alpha_i x_i$ with $(\textbf{units } \alpha_i) = \frac{(\textbf{common-unit})}{ (\textbf{units } x_i)}$. Now it makes sense to compare $x, y \in \mathcal{X}$ with a rescaled Euclidean distance, $\| \alpha \odot (x - y) \|_2$ or for, our purposes, $\rho(x) = \| \alpha \odot x \|^2_2$. For numerical-stability reasons, it's better to use the squared two-norm and pass in $\varepsilon^2$ which is, of course, mathematically equivalent.</p>
<p>The most notable thing about this example is that it demonstrates that the gradient is <em>covariant</em>: the conversion factor is <em>inverted</em> in the steepest-ascent direction.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">assert_symmetric_positive_definite</span><span class="p">(</span><span class="n">A</span><span class="p">):</span>
<span class="n">e</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigvals</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="k">assert</span> <span class="p">(</span><span class="n">e</span> <span class="o">></span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">all</span><span class="p">(),</span> <span class="n">e</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">random_spd</span><span class="p">(</span><span class="n">D</span><span class="p">):</span>
<span class="n">AA</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">D</span><span class="p">,</span><span class="n">D</span><span class="p">)</span>
<span class="n">U</span><span class="p">,</span><span class="n">S</span><span class="p">,</span><span class="n">Vt</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">AA</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">AA</span><span class="o">.</span><span class="n">T</span><span class="p">))</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="n">S</span> <span class="o">/=</span> <span class="n">S</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="k">return</span> <span class="n">U</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">S</span><span class="p">))</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Vt</span><span class="p">)</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">random_spd</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
<span class="n">assert_symmetric_positive_definite</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="k">if</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">d</span><span class="p">))</span>
<span class="c1">#p.Hessian = lambda d: 0.5/p(d) * (A + A.T) # Hessian of a weighted p-norms are undefined at zero!</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">d</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="n">sa</span> <span class="o">=</span> <span class="n">steepest</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Delta</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span> <span class="n">visualize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ga</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">nd</span><span class="o">.</span><span class="n">Gradient</span><span class="p">(</span><span class="n">f</span><span class="p">)(</span><span class="n">x0</span><span class="p">))</span> <span class="c1"># covariant! linear transform is inverted!</span>
<span class="n">sa</span> <span class="o">*=</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">norm</span><span class="p">(</span><span class="n">sa</span><span class="p">)</span>
<span class="n">ga</span> <span class="o">*=</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">norm</span><span class="p">(</span><span class="n">ga</span><span class="p">)</span>
<span class="n">compare</span><span class="p">(</span><span class="n">ga</span><span class="p">,</span> <span class="n">sa</span><span class="p">)</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
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<pre>100.0% (10000/10000) [================================================] 00:00:00
100.0% (10000/10000) [================================================] 00:00:00
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<pre>
Comparison: n=2
norms: <span class="ansi-green-fg">[1, 1]</span>
zero F1: <span class="ansi-green-fg">1</span>
pearson: <span class="ansi-green-fg">1</span>
spearman: <span class="ansi-green-fg">1</span>
Linf: <span class="ansi-red-fg">0.000143981</span>
same-sign: <span class="ansi-green-fg">100.0% (2/2)</span>
max rel err: <span class="ansi-green-fg">0.00026368</span>
regression: <span class="ansi-yellow-fg">[1.000 0.000]</span>
expect is larger: <span class="ansi-red-fg">100.0% (2/2)</span>
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<h2 id="Cheaper,-analytical-answers-with-Taylor-series-approximations">Cheaper, analytical answers with Taylor-series approximations<a class="anchor-link" href="#Cheaper,-analytical-answers-with-Taylor-series-approximations">¶</a></h2>
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<p>Since we've committed to studying changes of infinitesimal size, we can appeal to calculus to give us computationally cheap answers. Fundamentally, <em>derivatives</em> are less about <em>optimization</em> and more about <em>approximation</em>: "What will the response of the function $(\partial f)$ be to a small perturbation to its input $(\partial x)$?"</p>
<p>To derive a cheaper solution to our steepest ascent problem, we can leverage the infamous (truncated) <a href="https://en.wikipedia.org/wiki/Taylor_series">Taylor expansion</a> approximation. (The Taylor expansion is an amazing hammer, which I <a href="https://timvieira.github.io/blog/post/2014/07/21/expected-value-of-a-quadratic-and-the-delta-method/">wrote about</a> many years ago for approximating expectations of nonlinear functions.) In our case, this approximation can be used in both the objective and the constraints.</p>
<p><strong>Disclaimer</strong>: Note this is only a semi-precise analysis, It's enough to convince ourselves that a more precise analysis is likely to exist (with some carefully chosen stipulations). A more precise analysis would require taking limits and good stuff like that.</p>
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<p><strong>We'll start by approximating the objective...</strong></p>
<p>The (first-order) Taylor expansion of $f$ is a locally linear approximation to $f$,</p>
$$
\widehat{f_a}(x) \overset{\text{def}}{=} f(a) + \nabla f(a)^\top (x - a).
$$<p>So long as $|x-a|$ is small the approximation is fairly accurate</p>
$$
f(x) = \widehat{f_a}(x) + \mathcal{O}(|x-a|)
$$<p>Lucky for us we are in the small $|x-a|$ regime!</p>
<p>Our linearized steepest-direction problem is now</p>
$$
\begin{eqnarray}
d^*
&=& \underset{\rho(\Delta_d) = \varepsilon}{\textrm{argmax }} \widehat{f_a}(x + d) \\
&=& \underset{\rho(\Delta_d) = \varepsilon}{\textrm{argmax }} f(x) + \nabla f(x)^\top d \\
&=& \underset{\rho(\Delta_d) = \varepsilon}{\textrm{argmax }} \nabla f(x)^\top d.
\end{eqnarray}
$$
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<p><strong>Numerical Taylor approximation</strong></p>
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<div class="prompt input_prompt">In [18]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Visualize the quadratic approximation to the constraint boundary.</span>
<span class="c1">#Q = nd.Hessian(p)(0 * x0)</span>
<span class="c1">#phat = lambda d: 0.5 * d.T.dot(Q).dot(d)</span>
<span class="c1">#z = 1.75*max(np.abs(opt).max(), eps) </span>
<span class="c1">#X = Y = [-z, z, 100]</span>
<span class="c1">#contour_plot(lambda d: f(Delta(x0, d)), X, Y); pl.colorbar()</span>
<span class="c1">#contour_plot(lambda d: float(phat(d) <= eps), X, Y, color='Reds_r')</span>
<span class="c1">#contour_plot(lambda d: float(p(d) <= eps), X, Y, color='Blues_r')</span>
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<div class="prompt input_prompt">In [19]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># This version also works</span>
<span class="c1">#def steepest2(f, p, x0, **kw):</span>
<span class="c1"># "Numerical solution to the Taylor approximated objective."</span>
<span class="c1"># g = nd.Gradient(f)(x0) # linear approximation to objective</span>
<span class="c1"># Q = nd.Hessian(p)(x0 * 0)</span>
<span class="c1"># assert_symmetric_positive_definite(Q)</span>
<span class="c1"># fhat = lambda d: g.dot(d)</span>
<span class="c1"># phat = lambda d: 0.5 * d.T.dot(Q).dot(d) # quadratic approximation to constraint. </span>
<span class="c1"># return steepest(fhat, Delta, p, x0, **kw)</span>
<span class="k">def</span> <span class="nf">steepest3</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">):</span>
<span class="s2">"Analytic solution to the Taylor approximated objective."</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">nd</span><span class="o">.</span><span class="n">Gradient</span><span class="p">(</span><span class="n">f</span><span class="p">)(</span><span class="n">x0</span><span class="p">)</span>
<span class="n">Q</span> <span class="o">=</span> <span class="n">nd</span><span class="o">.</span><span class="n">Hessian</span><span class="p">(</span><span class="n">p</span><span class="p">)(</span><span class="mi">0</span> <span class="o">*</span> <span class="n">x0</span><span class="p">)</span>
<span class="n">assert_symmetric_positive_definite</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">Q</span> <span class="o">/</span> <span class="n">eps</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="k">return</span> <span class="n">x</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">sa</span> <span class="o">=</span> <span class="n">steepest</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Delta</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span> <span class="n">visualize</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">opt</span> <span class="o">=</span> <span class="n">steepest3</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">)</span>
<span class="c1">#opt = steepest2(f, p, x0, eps=eps)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="n">opt</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[</span><span class="n">opt</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">s</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'yellow'</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">1.</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="n">sa</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[</span><span class="n">sa</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'m'</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">1.</span><span class="p">)</span>
<span class="n">compare</span><span class="p">(</span><span class="n">sa</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span><span class="o">.</span><span class="n">show</span><span class="p">();</span>
<span class="c1">#assert np.allclose(sa, opt)</span>
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<pre>100.0% (10000/10000) [================================================] 00:00:00
100.0% (10000/10000) [================================================] 00:00:00
/home/timv/anaconda3/lib/python3.7/site-packages/scipy/stats/stats.py:3020: RuntimeWarning: invalid value encountered in double_scalars
prob = _betai(0.5*df, 0.5, df/(df+t_squared))
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Comparison: n=2
norms: <span class="ansi-green-fg">[0.00775687, 0.00775621]</span>
zero F1: <span class="ansi-green-fg">1</span>
pearson: <span class="ansi-green-fg">1</span>
spearman: <span class="ansi-green-fg">1</span>
Linf: <span class="ansi-red-fg">1.47653e-06</span>
same-sign: <span class="ansi-green-fg">100.0% (2/2)</span>
max rel err: <span class="ansi-green-fg">0.000348599</span>
regression: <span class="ansi-yellow-fg">[1.000 -0.000]</span>
got is larger: <span class="ansi-red-fg">100.0% (2/2)</span>
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TpH3NbMTgH+G/hbd9+Z9sPNHgQeBCgoKKCsrCyDkI9NTU1NLPvNJvmeY77m12bLFrrMmEHr7dup7NmTteeeS93WrTB9etKhxSJfv8cvZXt+GRUaM/uxu//qaMuOUQXQJ+V1b2B9A+tUmFlL4FSg6kjbmlkrQpF50d3/0tCHu/sEYAJAcXGxl5SUnEAq6ZWVlRHHfrNJvueYd/nt3x/GYT77DL7xDfjhD1m9fn1+5ZhG3n2P9WR7fplOJRmfZtm9J/jZc4B+ZlZkZq0Jg/ul9dYpTfnsW4B33N2j5eOiWWlFQD/go2j85llgibv/+wnGJ5JfVq2C3/42dFouLoaHHw43JhOJ2RGPaMzsDuC7QJGZpRaBDkDliXxwNObyI2Ay0AJ4zt0Xm9kTwFx3LyUUjReiwf4qQjEiWu8VwiB/LfCwux80s2HA3cAiM/sk+qj/5e4TTyRWkZy2d2+4Jmb+/NAE8957obAw6aikGTnaqbOZhFlcXYF/S1leDSw80Q+PCsDEesseT3m+F7i1gW2fBJ6st+wD0o/fiDRPS5fCm29CTQ1cdhmUlKgJpjS5IxYad/8C+MLM7gTWR//wY2YnE8ZFVsceoYgcu127Qn+yTz+FgoJwQ7KePZOOSpqpTGedvQIMTXl9EPgzcEmjRyQix889dFieNCkM/F99dTiSURNMSVCmhaalu+//8oW7748G8EUkW+zYEZpgrlgBvXuHJpjduiUdlUjGhWaLmY2JBugxs7HA1vjCEpGMuYe7XU6dGlrJjB4NgwapP5lkjUwLzUPAi2b2FOHCyArgntiiEpHMVFZCaSl88QX07Qs33gidOiUdlchhMio07v45cGl0xb25e3W8YYnIEdXVheth3n0XWrYMp8kuuEDtYyQrZdoZoAD4F6Cnu18bdU8e4u7PxhqdiHzdxo3w2muwYQN861tw3XXQoUPSUYk0KNNTZ78Hfgc8Fr1eDrxMuKBSRJpCbS289x588AGcfDLcdlsoNDqKkSyXaaHp6u6vmNmj8NVV/QdjjEtEUq1dG8ZitmyB888Pd71s1y7pqEQykmmh2WVmXYg6JJvZpcCO2KISkWD/fnjnHZg9O9wj5q674Kyzko5K5JhkWmj+jtDI8kwzmwF0IzS5FJG4fP55uKXy9u1huvLw4dCmTdJRiRyzTGedzTezK4H+hF5iy9z9QKyRiTRXe/aEJpgffwxdu8L994eW/iI5KtNZZ22BHwLDCKfP3jezp7/sfSYijWTJktAEc/duuPxyuPLKMH1ZJIdl+hv8B0LH5v+KXt8BvEADnZVF5BjV1MDEieGGZKefDnfeCT16JB2VSKPItND0d/fzU16/a2YL4ghIpFlxhwULYPJkOHAgjMMMHaommJJXMi00H5vZpe4+C8DMBgMz4gtLpBnYvj00wSwvhz59wtX9XbsmHZVIo8u00AwG7jGzNdHrbwBLzGwR4O5+XizRieQjd5gzJzTBhHBl/yWX6MJLyVuZFprRsUYh0lxs3RouvFyzJlwPc8MNcNppSUclEqtMC00/d5+ausDMxrv78zHEJJJ/Dh6EmTNh+vRwK+WbbgpX+OsoRpqBTAvN42b2HeAfgFOAZ4B9gAqNyNFs2BCOYjZsgAEDwqmyU05JOiqRJpNpobkS+Hvgk+j14+7+/+IJSSRP1NaGI5gZM0JfsttvD00wRZqZTAtNJ8KEgM+B3sAZZmbu7rFFJpLL1qwJrfwrK+HCC2HkyNBxWaQZyvRer7OASe4+GrgE6ImmN4t83b594cLL554L4zJ33x2mLavISDOW6RHNNcCVZva4uz9hZv8XKIwvLJEcVF4emmDu3AmDB4eLL1u3TjoqkcRlWmgeBeqAq4EnCO1o/o1wdCPSvO3ZA2+9Fa7w/7IJZp8+SUclkjUyvmDT3S8ys48B3H2bmem/atK8uR9qgrlnD1xxRXioCabIYTL9G3HAzFpw6MZn3QhHOCLNU3V1GItZsiQ0v7z77tAMU0S+JtNC82vgr0B3M3uScNOz/x1bVCLZyh0++SQ0waythREjYMgQOCnTeTUizU9Gfzvc/UXgJ8D/ATYAN7n7n0/0w81stJktM7NyM3skzfttzOzl6P3ZZlaY8t6j0fJlZjYq032KHLdt2+CFF8K05YIC+MEP4LLLVGREjiLjk8nuvhRY2lgfHJ2KewoYAVQAc8ys1N0/S1ntAWCbu59lZuOAXwC3m9kAYBxwDmGq9VQzOzva5mj7FDk2dXWHmmCawfXXQ3Gx2seIZCjJUctBQLm7rwQws5eAsUBqURgL/Cx6/irwGzOzaPlL7r4PWGVm5dH+yGCfIhlrtX07/O53sHYt9OsXmmCeemrSYYnklCQLTS9gbcrrCkL3gbTruHutme0AukTLZ9Xbtlf0/Gj7FDm6gwdhxgx6lpZC//5w881w7rk6ihE5DkkWmnR/Y+u3tGlonYaWpztZnrZNjpk9CDwIUFBQQFlZWYOBHq+amppY9ptN8jHH1lu30nXGDFpv20bV6aez9rzzqKuqCn3L8lA+fof15XuO2Z5fkoWmAki9qq03sL6BdSrMrCVwKlB1lG2Ptk8A3H0CMAGguLjYS0pKjiuJIykrKyOO/WaTvMrxwAEoK4PFi6F3b/j+91m9cWP+5NeAvPoOG5DvOWZ7fkkWmjlAPzMrAtYRBve/W2+dUmA88CFhSvU77u5mVgr8ycz+nTAZoB/wEeFI52j7FPm6L74IrfwrK+Gii0ITzLZtYePGpCMTyXmJFZpozOVHwGSgBfCcuy82syeAue5eCjwLvBAN9lcRCgfReq8QBvlrgYfd/SBAun02dW6SQ/btC7PJ5syBTp3gnnugb9+koxLJK4n2ynD3icDEesseT3m+F7i1gW2fBJ7MZJ8iaa1YEZpgVleHiy6vukpNMEVioKZM0vzs3h2aYC5cCN26wW23hTEZEYmFCo00H+5hoH/SpNAE88or4fLL1QRTJGb6GybNQ3U1vPEGLFsGPXuGsZiCgqSjEmkWVGgkv7nDxx/D22+HJpgjR8Kll6o/mUgTUqGR/LVtW5iyvGoVFBbCmDHQuXPSUYk0Oyo0kn/q6mD2bHjnnXDkcuON4doYtY8RSYQKjeSXzZtDG/916+Dss0MTzI4dk45KpFlToZH8cPAgvP9+eLRpA9/5DgwcqKMYkSygQiO5b926cBSzeXPosDx6NLRvn3RUIhJRoZHcdeAAvPsufPghdOgAd9wRWvqLSFZRoZHctHp1mFFWVRXudnnNNaEJpohkHRUayS1798KUKTBvXpiqPH48FBUlHZWIHIEKjeSOZcvC1f01NTB0aGiC2apV0lGJyFGo0Ej227UrNMFctCi0jRk3Dnr1Ovp2IpIVVGgke7nDp5+GJpj79oUjmGHDoEWLpCMTkWOgQiPZaefOcJps+fLQwn/MGOjePemoROQ4qNBIdnEPA/1TpoRWMqNGweDBaoIpksNUaCR7VFaGO16uXh1mko0ZE26vLCI5TYVGkldXB7NmhSaYLVqEAnPhhWofI5InVGgkWZs2hfYx69eHq/qvv15NMEXyjAqNJKO29lATzJNPhltvhQEDdBQjkodUaKTpVVSEo5gtW+C880ITzHbtko5KRGKiQiNNZ//+0ARz1qzQBPPOO6Ffv6SjEpGYqdBI01i5Mswo27YNLrkkNMFs0ybpqESkCajQSLz27oW334b586FLF7jvPjjjjKSjEpEmpEIj8Vm6FN58M/QqGzYMrrxSTTBFmiEVGml8NTWhP9nixXD66eGGZD17Jh2ViCREhUYajzssXBg6Le/fD1dfDZddpiaYIs1cIg2kzKyzmU0xsxXRz7R9RsxsfLTOCjMbn7L8YjNbZGblZvZrs3DxhZn90syWmtlCM/urmZ3WVDk1ezt2wJ/+BH/9K3TtCg89BFdcoSIjIskUGuARYJq79wOmRa8PY2adgZ8Cg4FBwE9TCtJvgQeBftFjdLR8CjDQ3c8DlgOPxpmEEI5i5syBp54KPcquvTYM+HfrlnRkIpIlkjp1NhYoiZ4/D5QB/1hvnVHAFHevAjCzKcBoMysDOrr7h9HyPwA3AZPc/e2U7WcBt8QUvwBUVnL6W2+FK/v79oUbb1QTTBH5mqQKTYG7bwBw9w1mlu5GI72AtSmvK6JlvaLn9ZfXdz/wcuOEK4epq4OZM6GsjNbbtoU7Xl5wgdrHiEhasRUaM5sKnJ7mrccy3UWaZX6E5amf/RhQC7x4hPgeJJx+o6CggLKysgzDylxNTU0s+01S66oqunzwAW2qqtj1jW+w5pprWLNjB0yfnnRoscjH77A+5Zj7sj2/2AqNu1/T0HtmtsnMekRHMz2AzWlWq+DQ6TWA3oRTbBXR89Tl61P2PR64ARju7ocVoHrxTQAmABQXF3tJSUlDqx63srIy4thvImpr4b33wq2Ve/aE730PBgxgSz7lmEZefYcNUI65L9vzS+rUWSkwHvh59PO1NOtMBv4lZQLASOBRd68ys2ozuxSYDdwD/BeAmY0mjPVc6e67Y86h+Vi7NjTB3Lo1nCIbNSqMy4iIZCCpQvNz4BUzewBYA9wKYGbFwEPu/r2ooPwzMCfa5okvJwYAPwB+D5wMTIoeAL8B2gBTohnPs9z9oSbIJz/t3w/TpsFHH4V7xNx1F5x1VtJRiUiOSaTQuHslMDzN8rnA91JePwc818B6A9Ms17+CjeXzz0MTzB07QhPM4cPVBFNEjos6A8jh9uyByZPhk0/ChZf33Qff+EbSUYlIDlOhkUOWLAlNMHfvhssvD00wW+pXREROjP4VkdAEc+JE+Oyz0ATzzjuhR4+koxKRPKFC05y5w4IF4VTZgQNhHGboUPUnE5FGpULTXG3fHgb7P/88jMGMGRPGZEREGpkKTXPjHqYrT5sWXl93XZhVpvYxIhITFZrmZOtWKC2FNWvC9TA33ACn6U4KIhIvFZrm4ODBr5pg0ro1fPvbcN55OooRkSahQpPvNmwI7WM2boQBA8KpslNOSToqEWlGVGjy1YEDoaPyzJnQrh3cfjt861tJRyUizZAKTT5asyYcxVRWwoUXwsiRaoIpIolRockn+/YdaoJ52mlw991w5plJRyUizZwKTb4oLw/XxezcCZdeCldfHQb+RUQSpkKT63bvDlf2L1gA3brB/fdDnz5JRyUi8hUVmlzlHnqTTZwYOi5fcUV4qAmmiGQZ/auUi6qrQ5flpUvDbZXvvjs0wxQRyUIqNLnEPdwnZvJkqK2FESNgyBA46aSkIxMRaZAKTa7Yti0M9q9cCWecEZpgdumSdFQiIkelQpPt6uoONcE86aTQn+zii9U+RkRyhgpNNtuyJVx4WVEB/fqFInPqqUlHJSJyTFRostHBg/DBB/Dee9CmDdx8M5x7ro5iRCQnqdBkm/Xrw1HMpk0wcCBcey20b590VCIix02FJlscOBDa+M+cGborjxsH3/xm0lGJiJwwFZpssHp1uCFZVVUY6B8xAtq2TToqEZFGoUKTpH37YMoUmDsXOnWC8eOhqCjpqEREGpUKTVKWL4c33ghX+Q8ZAlddpSaYIpKXVGia2u7d8NZbsHBhaIJ5223Qu3fSUYmIxEaFpqm4w+LFoQnm3r1QUgLDhqkJpojkvUSaZJlZZzObYmYrop+dGlhvfLTOCjMbn7L8YjNbZGblZvZrs8MvMDGzfzAzN7OuceeSkZ074aWX4NVXw1jM978fCo2KjIg0A0l1Y3wEmObu/YBp0evDmFln4KfAYGAQ8NOUgvRb4EGgX/QYnbJdH2AEsCbOBDLiDvPmwVNPhR5lI0fCAw9AQUHSkYmINJmk/ks9FiiJnj8PlAH/WG+dUcAUd68CMLMpwGgzKwM6uvuH0fI/ADcBk6Lt/gP4CfBafOFnoKqKgsmTwzTlwsLQBLNz50RDEhFJQlKFpsDdNwC4+wYz655mnV7A2pTXFdGyXtHz+ssxszHAOndfYEm1a6mrg9mz4Z13aFNZCQ89BBddpPYxItJsxVZozGwqkO5uXI9luos0y7yh5WbWLtr3yAzje5Bw+o2CggLKysoyDKthrbZto+uMGbTZupXdvXuzZsQI1lRXw/TpJ7zvbFVTU9Mof3bZKt/zA+WYD7I9v9gKjbtf09B7ZrbJzHpERzM9gM1pVqvg0Ok1gN6EU2wV0fPU5esaOJcdAAAHrElEQVSBM4Ei4Mujmd7AfDMb5O4b08Q3AZgAUFxc7CUlJfVXOSJ3Z3Xlbjbv3Ev39q0o/Gwe9umn0L17uPBy4EA2T5/Ose4315SVleV1jvmeHyjHfJDt+SV16qwUGA/8PPqZbjxlMvAvKRMARgKPunuVmVWb2aXAbOAe4L/cfRHw1Sk4M1sNFLv71sYO3t0pXbCelVt20X7rJs6cOZV9B6rpf81QTE0wRUQOk9Sss58DI8xsBWGG2M8BzKzYzJ4BiCYB/DMwJ3o88eXEAOAHwDNAOfA5hyYCNInVlbtZuWUXvRbO4dxJf6bl/v18OHgUq0tUZERE6kvkiMbdK4HhaZbPBb6X8vo54LkG1ht4lM8oPOFAG7B5514A9nboyKZ+57DmoqEcbN2GzTv3UtRVhUZEJJWuGDwO3TuGzsqVRf2pLOr/teUiInJIUqfOclphl3b07Xb4kUvfbu0p7NIuoYhERLKXjmiOg5kx5vyeh2addWxLYZd2JHbtjohIFlOhOU5mRlHX9hqTERE5Cp06ExGRWKnQiIhIrFRoREQkVio0IiISKxUaERGJlbl70jEkzsy2AF/EsOuuQKP3Wssy+Z5jvucHyjEfJJXfGe7e7WgrqdDEyMzmuntx0nHEKd9zzPf8QDnmg2zPT6fOREQkVio0IiISKxWaeE1IOoAmkO855nt+oBzzQVbnpzEaERGJlY5oREQkVio0x8HMOpvZFDNbEf3s1MB646N1VpjZ+JTlF5vZIjMrN7NfW722z2b2D2bmZtY17lzSiSs/M/ulmS01s4Vm9lczO62pckqJbbSZLYtieyTN+23M7OXo/dlmVpjy3qPR8mVmNirTfTalxs7PzPqY2btmtsTMFpvZj5sum/Ti+A6j91qY2cdm9kb8WRxZTL+np5nZq9HfwSVmNqRpsgHcXY9jfAD/CjwSPX8E+EWadToDK6OfnaLnnaL3PgKGAEa4DfW1Kdv1ASYTruvpmk/5ASOBltHzX6Tbb8x5tSDc+rsv0BpYAAyot84Pgaej5+OAl6PnA6L12wBF0X5aZLLPHM+vB3BRtE4HYHlS+cWVY8p2fwf8CXgjqfzizBF4Hvhe9Lw1cFpT5aQjmuMzlvClEf28Kc06o4Ap7l7l7tuAKcBoM+sBdHT3Dz1843+ot/1/AD8Bkhw8iyU/d3/b3Wuj7WcBveNMIo1BQLm7r3T3/cBLhFxTpeb+KjA8OiIbC7zk7vvcfRVQHu0vk302lUbPz903uPt8AHevBpYAvZogl4bE8R1iZr2B64FnmiCHo2n0HM2sI3AF8CyAu+939+1NkAugU2fHq8DdNwBEP7unWacXsDbldUW0rFf0vP5yzGwMsM7dF8QR9DGIJb967icc7TSlhmJOu05UFHcAXY6wbSb7bCpx5PeV6PTMhcDsRoz5WMWV438S/oNX1/ghH7M4cuwLbAF+F50efMbMmuxmWrrxWQPMbCpwepq3Hst0F2mWeUPLzaxdtO+RGe7/hDR1fvU++zGgFngxw89qLEeN7QjrNLQ83X/WkjoajSO/sJHZKcB/A3/r7juPO8IT1+g5mtkNwGZ3n2dmJScYX2OI43tsCVwE/I27zzazXxFOi//TiQSaKRWaBrj7NQ29Z2abzKyHu2+IThVtTrNaBVCS8ro3UBYt711v+XrgTMI51QXR2HlvYL6ZDXL3jSeQSloJ5PflvscDNwDDo1NrTamCMAaWNrZ661SYWUvgVKDqKNsebZ9NJZb8zKwVoci86O5/iSf0jMWR4xhgjJldB7QFOprZH939rnhSOKo4cqwAKtz9y6PRVwmFpmkkOeiVqw/glxw+WP6vadbpDKwiDJR3ip53jt6bA1zKocHy69Jsv5rkJgPEkh8wGvgM6JZQXi0JkxaKODTIek69dR7m8EHWV6Ln53D4IOtKwqDtUfeZ4/kZYZztP5PIqSlyrLdtCclPBoglR+B9oH/0/GfAL5ssp6R/cXLxQTgXOg1YEf388h/YYuCZlPXuJwzGlQP3pSwvBj4lzAj5DdGFs/U+YzXJFZpY8ovWWwt8Ej2eTiC36wgzpz4HHouWPQGMiZ63Bf4cxfoR0Ddl28ei7ZZx+EzBr+0zwd/NRs0PGEY49bIw5Xv72n+McjnHevsuIeFCE+Pv6QXA3Oi7/B+iWaJN8VBnABERiZVmnYmISKxUaEREJFYqNCIiEisVGhERiZUKjYiIxEqFRiTHmNm9ZtYz6ThEMqVCI5J77gVUaCRn6DoakSxgZv8E3Em4oHUrMA+YCjwNtCNcgHc/MBz4PbAO2AMMcfc9CYQskjEd0YgkzMyKge8QOiPfTOisAKH1yz+6+3nAIuCn7v4q4eruO939AhUZyQUqNCLJGwa85u57PNzz5XWgPeHGVNOjdZ4n3E9EJOeo0IgkL11rd5G8oUIjkrwPgBvNrG1035frgV3ANjO7PFrnbuDLo5tqwm2VRXKC7kcjkjB3n2NmpYT27l8QxmB2AOOBp6Ob4q0E7os2+X20XJMBJCdo1plIFjCzU9y9Jioq7wEPuvv8pOMSaQw6ohHJDhPMbADhPiPPq8hIPtERjYiIxEqTAUREJFYqNCIiEisVGhERiZUKjYiIxEqFRkREYqVCIyIisfr/pbdUT4ta4PUAAAAASUVORK5CYII=
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<h3 id="Application:-Steepest-ascent-under-different-p-norms">Application: Steepest ascent under different p-norms<a class="anchor-link" href="#Application:-Steepest-ascent-under-different-p-norms">¶</a></h3><p>We saw $L_2$ norms and weighted $L_2$ norms above. Here will see that $L_1$ and $L_\infty$ have pretty neat interpretations.
We have an optimization problem of the following form each $p$
$$
d^* = \underset{\|d\|_p = \varepsilon}{\textrm{argmax }} \nabla f(x)^\top d
$$</p>
<p>Each choice of <a href="https://en.wikipedia.org/wiki/Lp_space">$p$-norm</a> gives rise to a differently shaped constraint set and thus has different answers! We visualize each constraint set for the two-dimensional case for $p=1, 2, \text{ and }\infty$.</p>
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<h5 id="Drawing-constraint-boundaries">Drawing constraint boundaries<a class="anchor-link" href="#Drawing-constraint-boundaries">¶</a></h5>
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<div class="prompt input_prompt">In [21]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">eps</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="k">def</span> <span class="nf">draw_norm</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="n">zs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">ts</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">ts</span><span class="p">)))</span><span class="o">.</span><span class="n">T</span> <span class="c1"># unit sphere</span>
<span class="n">xs</span><span class="p">,</span> <span class="n">ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">z</span> <span class="o">*</span> <span class="n">eps</span> <span class="o">/</span> <span class="n">p</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="n">zs</span><span class="p">])</span><span class="o">.</span><span class="n">T</span> <span class="c1"># renormalize the unit sphere</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">)</span>
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">inf</span><span class="p">]:</span>
<span class="n">draw_norm</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="n">eps</span><span class="p">)</span>
</pre></div>
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<h3 id="$L_1$">$L_1$<a class="anchor-link" href="#$L_1$">¶</a></h3>
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<p>Maximizing the linearized objective under the $L_1$ polytope is a linear program. Unless the gradient is not parallel to the boundary of the polytope (i.e., a tie), we know that the optimum is at a corner! The corners, in this case, are one-hot vectors of length with $\pm \varepsilon$ as their single active value. We maximize the linearized objective by taking it's largest magnitude entry of the gradient and its sign,</p>
$$
k = \textrm{argmax}_i |g_i| \\
d^* = \varepsilon \cdot \textrm{onehot}(k, D) \cdot \textrm{sign}(g_k)
$$
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<div class="prompt input_prompt">In [28]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">steepest_l1_analytical</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="p">):</span>
<span class="n">x0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
<span class="n">g0</span> <span class="o">=</span> <span class="n">nd</span><span class="o">.</span><span class="n">Gradient</span><span class="p">(</span><span class="n">f</span><span class="p">)(</span><span class="n">x0</span><span class="p">)</span>
<span class="n">hottest</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">g0</span><span class="p">))</span>
<span class="k">return</span> <span class="n">onehot</span><span class="p">(</span><span class="n">hottest</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span> <span class="o">*</span> <span class="n">eps</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">g0</span><span class="p">[</span><span class="n">hottest</span><span class="p">])</span>
<span class="k">def</span> <span class="nf">steepest_l1_numerical</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="p">):</span>
<span class="c1"># An easier way to optimize L1 is to rotate the parameter space and use Linf (i.e., easy box constraints).</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">nd</span><span class="o">.</span><span class="n">Gradient</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
<span class="n">x0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
<span class="n">g0</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
<span class="c1"># Create a linear program for the linear maximization over the L1-polytope.</span>
<span class="n">A_ub</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span>
<span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span>
<span class="p">[</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="o">+</span><span class="mi">1</span><span class="p">],</span>
<span class="p">[</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">+</span><span class="mi">1</span><span class="p">],</span>
<span class="p">])</span>
<span class="n">b_ub</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span>
<span class="n">eps</span><span class="p">,</span>
<span class="n">eps</span><span class="p">,</span>
<span class="n">eps</span><span class="p">,</span>
<span class="n">eps</span><span class="p">,</span>
<span class="p">])</span>
<span class="n">opt</span> <span class="o">=</span> <span class="n">linprog</span><span class="p">(</span><span class="o">-</span><span class="n">g0</span><span class="p">,</span> <span class="n">A_ub</span><span class="o">=</span><span class="n">A_ub</span><span class="p">,</span> <span class="n">b_ub</span><span class="o">=</span><span class="n">b_ub</span><span class="p">,</span> <span class="n">bounds</span><span class="o">=</span><span class="p">[(</span><span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">)]</span><span class="o">*</span><span class="mi">2</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">opt</span><span class="o">.</span><span class="n">success</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">message</span>
<span class="k">return</span> <span class="n">opt</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="kn">import</span> <span class="n">linprog</span>
<span class="kn">from</span> <span class="nn">arsenal.viz</span> <span class="kn">import</span> <span class="n">contour_plot</span>
<span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">onehot</span>
<span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="mf">0.75</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">eps</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">norm</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">draw_norm</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">D</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">analytical</span> <span class="o">=</span> <span class="n">steepest_l1_analytical</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
<span class="n">numerical</span> <span class="o">=</span> <span class="n">steepest_l1_numerical</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span><span class="o">.</span><span class="n">x</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">analytical</span><span class="p">,</span> <span class="n">numerical</span><span class="p">)</span>
<span class="n">contour_plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">g0</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.25</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mf">1.25</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mi">100</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.25</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mf">1.25</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>
<span class="c1">#contour_plot(f, [-1.25*eps, 1.25*eps, 100], [-1.25*eps, 1.25*eps, 100])</span>
<span class="n">pl</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="n">analytical</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[</span><span class="n">analytical</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">);</span>
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<h3 id="$L_\infty$">$L_\infty$<a class="anchor-link" href="#$L_\infty$">¶</a></h3><p>Under similar conditions to "no ties," the gradient direction is maximized with a corner on the $\varepsilon$-unit box. The corners of the unit box are the sign function!</p>
<p>Therefore, steepest ascent in $L_\infty$ is just the sign of the gradient!</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Analytical solution is simple: just the sign of the gradient!</span>
<span class="n">analytical</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">g0</span><span class="p">)</span> <span class="o">*</span> <span class="n">eps</span>
<span class="c1"># visualize the norm constraint</span>
<span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="n">draw_norm</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'k'</span><span class="p">)</span>
<span class="c1"># The L-inf norm is an easy box-constrained problem.</span>
<span class="n">opt</span> <span class="o">=</span> <span class="n">minimize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="o">-</span><span class="n">g0</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="o">*</span><span class="n">x0</span><span class="p">,</span> <span class="n">bounds</span><span class="o">=</span><span class="p">[(</span><span class="o">-</span><span class="n">eps</span><span class="p">,</span> <span class="n">eps</span><span class="p">)]</span><span class="o">*</span><span class="mi">2</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">opt</span><span class="o">.</span><span class="n">success</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">message</span>
<span class="c1"># Make sure we got what was expected!</span>
<span class="n">numerical</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">x</span>
<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numerical</span><span class="p">,</span> <span class="n">analytical</span><span class="p">)</span>
<span class="n">contour_plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">g0</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.25</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mf">1.25</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mi">100</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.25</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mf">1.25</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>
<span class="n">pl</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="n">opt</span><span class="o">.</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[</span><span class="n">opt</span><span class="o">.</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">);</span>
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<pre>100.0% (10000/10000) [================================================] 00:00:00
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<h2 id="Multiplicative-updates">Multiplicative updates<a class="anchor-link" href="#Multiplicative-updates">¶</a></h2>
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<p>Just for fun, let's work out an example of a multiplicative update.</p>
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<div class="prompt input_prompt">In [34]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">arsenal.maths</span> <span class="kn">import</span> <span class="n">entropy</span><span class="p">,</span> <span class="n">kl_divergence</span><span class="p">,</span> <span class="n">softmax</span>
<span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="kn">import</span> <span class="n">minimize</span>
<span class="kn">from</span> <span class="nn">arsenal.viz</span> <span class="kn">import</span> <span class="n">contour_plot</span>
<span class="k">def</span> <span class="nf">Delta</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span> <span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="n">d</span>
<span class="k">def</span> <span class="nf">steepest_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">eps</span><span class="p">):</span>
<span class="n">opt</span> <span class="o">=</span> <span class="n">minimize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="o">-</span><span class="n">f</span><span class="p">(</span><span class="n">Delta</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">d</span><span class="p">)),</span>
<span class="n">x0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="c1"># multiplicative updates should start at one!</span>
<span class="n">options</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">maxiter</span> <span class="o">=</span> <span class="mi">5000</span><span class="p">),</span>
<span class="n">bounds</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="kc">None</span><span class="p">)]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
<span class="n">constraints</span> <span class="o">=</span> <span class="p">[{</span><span class="s1">'fun'</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">eps</span> <span class="o">-</span> <span class="n">p</span><span class="p">(</span><span class="n">d</span><span class="p">),</span> <span class="s1">'type'</span><span class="p">:</span> <span class="s1">'ineq'</span><span class="p">}])</span>
<span class="k">assert</span> <span class="n">opt</span><span class="o">.</span><span class="n">success</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">message</span>
<span class="k">return</span> <span class="n">opt</span>
<span class="n">D</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span> <span class="n">x0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">.4</span><span class="p">,</span> <span class="mf">.6</span><span class="p">]);</span> <span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span> <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">d</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="n">eps</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="c1"># crank-up epsilon to see that the constraint boundary is nonconvex.</span>
<span class="n">opt</span> <span class="o">=</span> <span class="n">steepest_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">eps</span><span class="p">),</span> <span class="mi">1</span><span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mi">100</span><span class="p">]</span>
<span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">eps</span><span class="p">),</span> <span class="mi">1</span><span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mi">100</span><span class="p">]</span>
<span class="n">pl</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">contour_plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">f</span><span class="p">(</span><span class="n">Delta</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">d</span><span class="p">)),</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
<span class="n">contour_plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="nb">float</span><span class="p">(</span><span class="n">p</span><span class="p">(</span><span class="n">d</span><span class="p">)</span> <span class="o"><=</span> <span class="n">eps</span><span class="p">),</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'binary_r'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="n">opt</span><span class="o">.</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[</span><span class="n">opt</span><span class="o">.</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">);</span>
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<pre>/home/timv/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:18: RuntimeWarning: divide by zero encountered in log
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<p>Multiplicative updates are much easier to work with geometrically if we switch to log-space. This is because because p goes from L1-of-log (nonconvex) -> L1 (convex). This whole thing is equivalent to additive steepest-ascent in log space. So here we see our friend L1 again.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">eps</span> <span class="o">=</span> <span class="mf">.001</span>
<span class="n">X</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.5</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mf">1.5</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mi">100</span><span class="p">]</span>
<span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.5</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mf">1.5</span><span class="o">*</span><span class="n">eps</span><span class="p">,</span> <span class="mi">100</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">Delta</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span> <span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">steepest_mul_logspace</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">eps</span><span class="p">):</span>
<span class="n">opt</span> <span class="o">=</span> <span class="n">minimize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="o">-</span><span class="n">f</span><span class="p">(</span><span class="n">Delta</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">d</span><span class="p">)),</span>
<span class="n">x0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
<span class="n">options</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">maxiter</span> <span class="o">=</span> <span class="mi">10000</span><span class="p">),</span>
<span class="n">constraints</span> <span class="o">=</span> <span class="p">[{</span><span class="s1">'fun'</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">eps</span> <span class="o">-</span> <span class="n">p</span><span class="p">(</span><span class="n">d</span><span class="p">),</span> <span class="s1">'type'</span><span class="p">:</span> <span class="s1">'ineq'</span><span class="p">}])</span>
<span class="k">assert</span> <span class="n">opt</span><span class="o">.</span><span class="n">success</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">message</span>
<span class="k">return</span> <span class="n">opt</span>
<span class="n">p</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">d</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="n">opt</span> <span class="o">=</span> <span class="n">steepest_mul_logspace</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
<span class="n">contour_plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="n">f</span><span class="p">(</span><span class="n">Delta</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">d</span><span class="p">)),</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">);</span> <span class="n">pl</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
<span class="n">contour_plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">d</span><span class="p">:</span> <span class="nb">float</span><span class="p">(</span><span class="n">p</span><span class="p">(</span><span class="n">d</span><span class="p">)</span> <span class="o"><=</span> <span class="n">eps</span><span class="p">),</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'binary_r'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="n">opt</span><span class="o">.</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[</span><span class="n">opt</span><span class="o">.</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">);</span>
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<pre>100.0% (10000/10000) [================================================] 00:00:00
100.0% (10000/10000) [================================================] 00:00:00
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<h2 id="Summary">Summary<a class="anchor-link" href="#Summary">¶</a></h2>
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<p>In this post, we talked about steepest ascent in a number of different space (i.e., under different metrics). We even touched on the idea of non-additive changes. We saw that under the $L_1$ and $L_\infty$ metrics we get some really cute interpretations of what the steepest direction is!</p>
<p>Further reading:</p>
<ul>
<li>Marc Toussaint, <a href="https://ipvs.informatik.uni-stuttgart.de/mlr/marc/notes/gradientDescent.pdf">Some notes on gradient descent</a>.</li>
</ul>
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</script>Black-box optimization2018-03-16T00:00:00-04:002018-03-16T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2018-03-16:/blog/post/2018/03/16/black-box-optimization/<p>Black-box optimization algorithms are a fantastic tool that everyone should be
aware of. I frequently use black-box optimization algorithms for prototyping and
when gradient-based algorithms fail,
e.g., because the function is not differentiable,
because the function is truly opaque (no gradients),
because the gradient would require too much memory to compute efficiently.</p>
<p>From a young age, we are taught to love gradients and never learn about any
optimization algorithms other than gradient descent. I believe this obsession
has put us in a local optimum. I've been amazed at how few people know about
non-gradient algorithms for optimization. Although, this is slowly improving
thanks to the prevalence of hyperparameter optimization, so most people have
used random search and at least know of Bayesian optimization.</p>
<p>There are many ways to optimize a function! The gradient just happens to have a
<a href="/blog/post/2017/08/18/backprop-is-not-just-the-chain-rule/">beautiful</a> and
<a href="/blog/post/2016/09/25/evaluating-fx-is-as-fast-as-fx/">computationally efficient</a>
shortcut for finding <em>the direction of steepest descent</em> in Euclidean space.</p>
<p><strong>What is a descent direction anyway?</strong> For minimizing an function <span class="math">\(f:
\mathbb{R}^d \mapsto \mathbb{R}\)</span>, a descent direction for <span class="math">\(f\)</span> is a
<span class="math">\((d+1)\)</span>-dimensional hyperplane. The gradient gives a unique hyperplane that is
tangent to the surface of <span class="math">\(f\)</span> at the point <span class="math">\(x\)</span>; the <span class="math">\((d+1)^{\text{th}}\)</span>
coordinate comes from the value <span class="math">\(f(x)\)</span>—think of it like a first-order
Taylor approximation to <span class="math">\(f\)</span> at <span class="math">\(x\)</span>. (For an in-depth discussion on notions of
<em>steepest</em> descent, check out
<a href="https://timvieira.github.io/blog/post/2019/04/19/steepest-ascent/">this post</a>.)</p>
<p><strong>The baseline:</strong> Without access to gradient code, <em>approximating</em> the gradient
takes <span class="math">\(d+1\)</span> function evaluations via the finite-difference approximation to the
gradient,<sup id="sf-black-box-optimization-1-back"><a href="#sf-black-box-optimization-1" class="simple-footnote" title="Of course, it's better to use the two-sided difference approximation to the gradient in practice, which requires \(2 \cdot d\) function evaluations, not \(d+1\). ">1</a></sup> which I've discussed a
<a href="http://timvieira.github.io/blog/post/2014/02/10/gradient-vector-product/">few</a>
<a href="http://timvieira.github.io/blog/post/2017/04/21/how-to-test-gradient-implementations/">times</a>. This
shouldn't be surprising since that's the size of the object we're looking for
anyways!<sup id="sf-black-box-optimization-2-back"><a href="#sf-black-box-optimization-2" class="simple-footnote" title="Note that we can get noisy, approximations with much fewer than \(\mathcal{O}(d)\) evaluations, e.g., SPSA or even REINFORCE obtain gradients approximations with just \(\mathcal{O}(1)\) evaluations per iteration.">2</a></sup></p>
<p><strong>Can we do better?</strong> Suppose we had <span class="math">\((d+1)\)</span> arbitrary points
<span class="math">\(\boldsymbol{x}^{(1)}, \ldots, \boldsymbol{x}^{(d+1)}\)</span> in <span class="math">\(\mathbb{R}^n\)</span> with
values <span class="math">\(f^{(i)} = f(\boldsymbol{x}^{(i)}).\)</span> Can we find efficiently find a
descent direction without extra <span class="math">\(f\)</span> evaluations?</p>
<p><strong>The Nelder-Mead trick:</strong> Take the worst-performing point in this set
<span class="math">\(\boldsymbol{x}^{(\text{worst})}\)</span> and consider moving that point through the
center-of-mass of the <span class="math">\(d\)</span> remaining points. Call this the NM direction. At some
point along that direction (think line search) there will be a good place to put
that point, which will make it the new best point. We can now repeat this
process: pick the worst point, reflect it through the center of mass, etc.</p>
<ul>
<li>
<p>The cost of finding the NM descent direction requires no additional function
evaluations, which allows the method to be very frugal with function
evaluations. Of course, stepping in the search direction should use line
search, which will require additional function evaluations; gradient-based
methods also benefit from line search.</p>
</li>
<li>
<p>Finding the worst point can be done in time <span class="math">\(\mathcal{O}(\log d)\)</span> using a
<a href="https://en.wikipedia.org/wiki/Heap_(data_structure)">heap</a>.</p>
</li>
<li>
<p>This NM direction might is not the steepest descent direction—like the
gradient—but it does give a reasonable descent direction to
follow. Often, the NM direction is more useful than the gradient direction
because it is not based on an infinitesimal ball around the current point
like the gradient. NM often "works" on noisy and nonsmooth functions where
gradients do not exist.</p>
</li>
<li>
<p>On high-dimensional problems, NM requires a significant number of "warm up"
<em>function</em> evaluations before it can take its first informed step. Whereas,
gradient descent could plausibly CONVERGE in fewer <em>gradient</em> evaluations
(assuming sufficiently "nice" functions)! So, if you have high-dimensional
problem and efficient gradients, use them.</p>
</li>
<li>
<p>In three dimensions, we can visualize this as a tetrahedron with corners that
"stick" to the surface of the function. At each iterations, the highest
(i.e., worst performing) point is the one most likely to be affected by
"gravity" which causes it to flip through the middle of the blob, as the
other points stay stuck.</p>
</li>
</ul>
<p></p><center>
<img alt="Nelder-Mead animation" src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/de/Nelder-Mead_Himmelblau.gif/320px-Nelder-Mead_Himmelblau.gif">
<br><em>(animation source: Wikipedia page for Nelder-Mead)</em>
</center><p></p>
<ul>
<li>This is exactly the descent direction used in the
<a href="https://en.wikipedia.org/wiki/Nelder%E2%80%93Mead_method">Nelder-Mead algorithm</a>
(Nelder & Mead, 1965), which happens to be a great default algorithm for
locally optimizing functions without access to gradients. Matlab and scipy
users may know it better as
<a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin.html"><code>fmin</code></a>.
There are some additional "search moves" required to turn NM into a robust
algorithm; these include shrinking and growing the set of points. I won't try
to make yet another tutorial on the specifics of Nelder-Mead, as several
already exist, but rather bring it to your attention as a plausible approach
for efficiently finding descent directions. You can find a tutorial with
plenty of visualization on its
<a href="https://en.wikipedia.org/wiki/Nelder%E2%80%93Mead_method">Wikipedia page</a>.</li>
</ul>
<h3>Summary</h3>
<p>I described the Nelder-Mead search direction as an efficient way to leverage
past function evaluations to find a descent directions, which serves as a
reasonable alternative to gradients when they are unavailable (or not useful).</p>
<h3>Further reading</h3>
<ul>
<li>There are plenty of other black-box optimization algorithms out there. The
wiki page on
<a href="https://en.wikipedia.org/wiki/Derivative-free_optimization">derivative-free optimization</a>
is a good starting point for learning more.</li>
</ul>
<h3>Footnotes</h3>
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</script><ol class="simple-footnotes"><li id="sf-black-box-optimization-1">Of course, it's better to use the two-sided difference
approximation to the gradient in practice, which requires <span class="math">\(2 \cdot d\)</span> function
evaluations, not <span class="math">\(d+1\)</span>.
<a href="#sf-black-box-optimization-1-back" class="simple-footnote-back">↩</a></li><li id="sf-black-box-optimization-2">Note that we can get noisy, approximations with much fewer
than <span class="math">\(\mathcal{O}(d)\)</span> evaluations, e.g.,
<a href="https://en.wikipedia.org/wiki/Simultaneous_perturbation_stochastic_approximation">SPSA</a>
or even REINFORCE obtain gradients approximations with just <span class="math">\(\mathcal{O}(1)\)</span>
evaluations per iteration. <a href="#sf-black-box-optimization-2-back" class="simple-footnote-back">↩</a></li></ol>Backprop is not just the chain rule2017-08-18T00:00:00-04:002017-08-18T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2017-08-18:/blog/post/2017/08/18/backprop-is-not-just-the-chain-rule/<p>Almost everyone I know says that "backprop is just the chain rule." Although
that's <em>basically true</em>, there are some subtle and beautiful things about
automatic differentiation techniques (including backprop) that will not be
appreciated with this <em>dismissive</em> attitude.</p>
<p>This leads to a poor understanding. As
<a href="http://timvieira.github.io/blog/post/2016/09/25/evaluating-fx-is-as-fast-as-fx/">I have ranted before</a>:
people do not understand basic facts about autodiff.</p>
<ul>
<li>Evaluating <span class="math">\(\nabla f(x)\)</span> is provably as fast as evaluating <span class="math">\(f(x)\)</span>.</li>
</ul>
<!-- Let that sink in. Computing the gradient—an essential ingredient to
efficient optimization—is no slower to compute than the function
itself. Contrast that with the finite-difference gradient approximation, which
is quite accurate, but its runtime is $\textrm{dim}(x)$ times slower than
evaluating $f$
([discussed here](http://timvieira.github.io/blog/post/2017/04/21/how-to-test-gradient-implementations/))!
-->
<ul>
<li>Code for <span class="math">\(\nabla f(x)\)</span> can be derived by a rote program transformation, even
if the code has control flow structures like loops and intermediate variables
(as long as the control flow is independent of <span class="math">\(x\)</span>). You can even do this
"automatic" transformation by hand!</li>
</ul>
<h3>Autodiff <span class="math">\(\ne\)</span> what you learned in calculus</h3>
<p>Let's try to understand the difference between autodiff and the type of
differentiation that you learned in calculus, which is called <em>symbolic</em>
differentiation.</p>
<p>I'm going to use an example from
<a href="https://people.cs.umass.edu/~domke/courses/sml2011/08autodiff_nnets.pdf">Justin Domke's notes</a>,
</p>
<div class="math">$$
f(x) = \exp(\exp(x) + \exp(x)^2) + \sin(\exp(x) + \exp(x)^2).
$$</div>
<!--
If we plug-and-chug with the chain rule, we get a correct expression for the
derivative,
$$\small
\frac{\partial f}{\partial x} =
\exp(\exp(x) + \exp(x)^2) (\exp(x) + 2 \exp(x) \exp(x)) \\
\quad\quad\small+ \cos(\exp(x) + \exp(x)^2) (\exp(x) + 2 \exp(x) \exp(x)).
$$
However, this expression leaves something to be desired because it has a lot of
repeated evaluations of the same function. This is clearly bad, if we want to
turn it into source code.
-->
<p>If we were writing <em>a program</em> (e.g., in Python) to compute <span class="math">\(f\)</span>, we'd take
advantage of the fact that it has a lot of repeated evaluations for efficiency.</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="n">e</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="k">return</span> <span class="n">d</span> <span class="o">+</span> <span class="n">e</span>
</code></pre></div>
<p>Symbolic differentiation would have to use the "flat" version of this function,
so no intermediate variable <span class="math">\(\Rightarrow\)</span> slow.</p>
<p>Automatic differentiation lets us differentiate a program with <em>intermediate</em>
variables.</p>
<ul>
<li>
<p>The rules for transforming the code for a function into code for the gradient
are really minimal (fewer things to memorize!). Additionally, the rules are
more general than in symbolic case because they handle as a superset of
programs.</p>
</li>
<li>
<p>Quite <a href="http://conal.net/papers/beautiful-differentiation/">beautifully</a>, the
program for the gradient <em>has exactly the same structure</em> as the function,
which implies that we get the same runtime (up to some constants factors).</p>
</li>
</ul>
<p>I won't give the details of how to execute the backpropagation transform to the
program. You can get that from
<a href="https://people.cs.umass.edu/~domke/courses/sml2011/08autodiff_nnets.pdf">Justin Domke's notes</a>
and many other good
resources. <a href="https://gist.github.com/timvieira/39e27756e1226c2dbd6c36e83b648ec2">Here's some code</a>
that I wrote that accompanies to the <code>f(x)</code> example, which has a bunch of
comments describing the manual "automatic" differentiation process on <code>f(x)</code>.</p>
<!--
Caveat: You might have seen some *limited* cases where an input variable was
reused, but chances are that it was something really simple like multiplication
or division, e.g., $\nabla\! \left[ f(x) \cdot g(x) \right] = f(x) \cdot g'(x)
+ f'(x) \cdot g(x)$, and you just memorized a rule. The rules of autodiff are
simpler and actually explains why there is a sum in the product rule. You can
also rederive the quotient rule without a hitch. I'm all about having fewer
things to memorize!
-->
<!--
You might hope that something like common subexpression elimination would save
the symbolic approach. Indeed that could be leveraged to improve any chunk of
code, but to match efficiency it's not needed! If we had needed to blow up the
computation to then shrink it down that would be much less efficient! The "flat"
version of a program can be exponentially larger than a version with reuse.
-->
<!-- Only sort of related: think of the exponential blow up in converting a
Boolean expression from conjunctive normal form to and disjunction normal. -->
<h2>Autodiff by the method of Lagrange multipliers</h2>
<p>Let's view the intermediate variables in our optimization problem as simple
equality constraints in an equivalent <em>constrained</em> optimization problem. It
turns out that the de facto method for handling constraints, the method Lagrange
multipliers, recovers <em>exactly</em> the adjoints (intermediate derivatives) in the
backprop algorithm!</p>
<p>Here's our example from earlier written in this constraint form:</p>
<div class="math">$$
\begin{align*}
\underset{x}{\text{argmax}}\ & f \\
\text{s.t.} \quad
a &= \exp(x) \\
b &= a^2 \\
c &= a + b \\
d &= \exp(c) \\
e &= \sin(c) \\
f &= d + e
\end{align*}
$$</div>
<h4>The general formulation</h4>
<div class="math">\begin{align*}
& \underset{\boldsymbol{x}}{\text{argmax}}\ z_n & \\
& \text{s.t.}\quad z_i = x_i &\text{ for $1 \le i \le d$} \\
& \phantom{\text{s.t.}}\quad z_i = f_i(z_{\alpha(i)}) &\text{ for $d < i \le n$} \\
\end{align*}</div>
<p>The first set of constraint (<span class="math">\(1, \ldots, d\)</span>) are a little silly. They are only
there to keep our formulation tidy. The variables in the program fall into three
categories:</p>
<ul>
<li>
<p><strong>input variables</strong> (<span class="math">\(\boldsymbol{x}\)</span>): <span class="math">\(x_1, \ldots, x_d\)</span></p>
</li>
<li>
<p><strong>intermediate variables</strong>: (<span class="math">\(\boldsymbol{z}\)</span>): <span class="math">\(z_i = f_i(z_{\alpha(i)})\)</span> for
<span class="math">\(1 \le i \le n\)</span>, where <span class="math">\(\alpha(i)\)</span> is a list of indices from <span class="math">\(\{1, \ldots,
n-1\}\)</span> and <span class="math">\(z_{\alpha(i)}\)</span> is the subvector of variables needed to evaluate
<span class="math">\(f_i(\cdot)\)</span>. Minor detail: take <span class="math">\(f_{1:d}\)</span> to be the identity function.</p>
</li>
<li>
<p><strong>output variable</strong> (<span class="math">\(z_n\)</span>): We assume that our programs has a singled scalar
output variable, <span class="math">\(z_n\)</span>, which represents the quantity we'd like to maximize.</p>
</li>
</ul>
<!-- (It is possible to generalize this story to compute Jacobians of functions
with multivariate outputs by "scalarizing" the objective, e.g., multiply the
outputs by a vector. This Gives an efficient program for computing Jacobian
vector products that can be used to extra Jacobians.) -->
<p>The relation <span class="math">\(\alpha\)</span> is a
<a href="https://en.wikipedia.org/wiki/Dependency_graph">dependency graph</a> among
variables. Thus, <span class="math">\(\alpha(i)\)</span> is the list of <em>incoming</em> edges to node <span class="math">\(i\)</span> and
<span class="math">\(\beta(j) = \{ i: j \in \alpha(i) \}\)</span> is the set of <em>outgoing</em> edges. For now,
we'll assume that the dependency graph given by <span class="math">\(\alpha\)</span> is ① acyclic: no <span class="math">\(z_i\)</span>
can transitively depend on itself. ② single-assignment: each <span class="math">\(z_i\)</span> appears on
the left-hand side of <em>exactly one</em> equation. We'll discuss relaxing these
assumptions in <a href="#lagrange-backprop-generalization">§ Generalizations</a>.</p>
<p>The standard way to solve a constrained optimization is to use the method
Lagrange multipliers, which converts a <em>constrained</em> optimization problem into
an <em>unconstrained</em> problem with a few more variables <span class="math">\(\boldsymbol{\lambda}\)</span> (one
per <span class="math">\(x_i\)</span> constraint), called Lagrange multipliers.</p>
<h4>The Lagrangian</h4>
<p>To handle constraints, let's dig up a tool from our calculus class,
<a href="https://en.wikipedia.org/wiki/Lagrange_multiplier">the method of Lagrange multipliers</a>,
which converts a <em>constrained</em> optimization problem into an <em>unconstrained</em>
one. The unconstrained version is called "the Lagrangian" of the constrained
problem. Here is its form for our task,</p>
<div class="math">$$
\mathcal{L}\left(\boldsymbol{x}, \boldsymbol{z}, \boldsymbol{\lambda}\right)
= z_n - \sum_{i=1}^n \lambda_i \cdot \left( z_i - f_i(z_{\alpha(i)}) \right).
$$</div>
<p>Optimizing the Lagrangian amounts to solving the following nonlinear system of
equations, which give necessary, but not sufficient, conditions for optimality,</p>
<div class="math">$$
\nabla \mathcal{L}\left(\boldsymbol{x}, \boldsymbol{z}, \boldsymbol{\lambda}\right) = 0.
$$</div>
<p>Let's look a little closer at the Lagrangian conditions by breaking up the
system of equations into salient parts, corresponding to which variable types
are affected.</p>
<p><strong>Intermediate variables</strong> (<span class="math">\(\boldsymbol{z}\)</span>): Optimizing the
multipliers—i.e., setting the gradient of Lagrangian
w.r.t. <span class="math">\(\boldsymbol{\lambda}\)</span> to zero—ensures that the constraints on
intermediate variables are satisfied.</p>
<div class="math">$$
\begin{eqnarray*}
\nabla_{\! \lambda_i} \mathcal{L}
= z_i - f_i(z_{\alpha(i)}) = 0
\quad\Leftrightarrow\quad z_i = f_i(z_{\alpha(i)})
\end{eqnarray*}
$$</div>
<p>We can use forward propagation to satisfy these equations, which we may regard
as a block-coordinate step in the context of optimizing the <span class="math">\(\mathcal{L}\)</span>.</p>
<!--GENERALIZATION:
However, if they are cyclic dependencies we may need to
solve a nonlinear system of equations. (TODO: it's unclear what the more general
cyclic setting is. Perhaps I should having a running example of a cyclic program
and an acyclic program.)
-->
<p><strong>Lagrange multipliers</strong> (<span class="math">\(\boldsymbol{\lambda}\)</span>, excluding <span class="math">\(\lambda_n\)</span>):
Setting the gradient of the <span class="math">\(\mathcal{L}\)</span> w.r.t. the intermediate variables
equal to zeros tells us what to do with the intermediate multipliers.</p>
<div class="math">\begin{eqnarray*}
0 &=& \nabla_{\! z_j} \mathcal{L} \\
&=& \nabla_{\! z_j}\! \left[ z_n - \sum_{i=1}^n \lambda_i \cdot \left( z_i - f_i(z_{\alpha(i)}) \right) \right] \\
&=& - \sum_{i=1}^n \lambda_i \nabla_{\! z_j}\! \left[ \left( z_i - f_i(z_{\alpha(i)}) \right) \right] \\
&=& - \left( \sum_{i=1}^n \lambda_i \nabla_{\! z_j}\! \left[ z_i \right] \right) + \left( \sum_{i=1}^n \lambda_i \nabla_{\! z_j}\! \left[ f_i(z_{\alpha(i)}) \right] \right) \\
&=& - \lambda_j + \sum_{i \in \beta(j)} \lambda_i \frac{\partial f_i(z_{\alpha(i)})}{\partial z_j} \\
&\Updownarrow& \\
\lambda_j &=& \sum_{i \in \beta(j)} \lambda_i \frac{\partial f_i(z_{\alpha(i)})}{\partial z_j} \\
\end{eqnarray*}</div>
<p>Clearly, <span class="math">\(\frac{\partial f_i(z_{\alpha(i)})}{\partial z_j} = 0\)</span> for <span class="math">\(j \notin
\alpha(i)\)</span>, which is why the <span class="math">\(\beta(j)\)</span> notation came in handy. By assumption,
the local derivatives, <span class="math">\(\frac{\partial f_i(z_{\alpha(i)})}{\partial z_j}\)</span> for <span class="math">\(j
\in \alpha(i)\)</span>, are easy to calculate—we don't even need the chain rule to
compute them because they are simple function applications without
composition. Similar to the equations for <span class="math">\(\boldsymbol{z}\)</span>, solving this linear
system is another block-coordinate step.</p>
<p><em>Key observation</em>: The last equation for <span class="math">\(\lambda_j\)</span> should look very familiar:
It is exactly the equation used in backpropagation! It says that we sum
<span class="math">\(\lambda_i\)</span> of nodes that immediately depend on <span class="math">\(j\)</span> where we scaled each
<span class="math">\(\lambda_i\)</span> by the derivative of the function that directly relates <span class="math">\(i\)</span> and
<span class="math">\(j\)</span>. You should think of the scaling as a "unit conversion" from derivatives of
type <span class="math">\(i\)</span> to derivatives of type <span class="math">\(j\)</span>.</p>
<p><strong>Output multiplier</strong> (<span class="math">\(\lambda_n\)</span>): Here we follow the same pattern as for
intermediate multipliers.</p>
<div class="math">$$
\begin{eqnarray*}
0 &=& \nabla_{\! z_n}\! \left[ z_n - \sum_{i=1}^n \lambda_i \cdot \left( z_i - f_i(z_{\alpha(i)}) \right) \right] &=& 1 - \lambda_n \\
&\Updownarrow& \\
\lambda_n &=& 1
\end{eqnarray*}
$$</div>
<p><strong>Input multipliers</strong> <span class="math">\((\boldsymbol{\lambda}_{1:d})\)</span>: Our dummy constraints
gives us <span class="math">\(\boldsymbol{\lambda}_{1:d}\)</span>, which are conveniently equal to the
gradient of the function we're optimizing:</p>
<div class="math">$$
\nabla_{\!\boldsymbol{x}} f(\boldsymbol{x}) = \boldsymbol{\lambda}_{1:d}.
$$</div>
<p>Of course, this interpretation is only precise when ① the constraints are
satisfied (<span class="math">\(\boldsymbol{z}\)</span> equations) and ② the linear system on multipliers is
satisfied (<span class="math">\(\boldsymbol{\lambda}\)</span> equations).</p>
<p><strong>Input variables</strong> (<span class="math">\(\boldsymbol{x}\)</span>): Unfortunately, the there is no
closed-form solution to how to set <span class="math">\(\boldsymbol{x}\)</span>. For this we resort to
something like gradient ascent. Conveniently, <span class="math">\(\nabla_{\!\boldsymbol{x}}
f(\boldsymbol{x}) = \boldsymbol{\lambda}_{1:d}\)</span>, which we can use to optimize
<span class="math">\(\boldsymbol{x}\)</span>!</p>
<div id="lagrange-backprop-generalization"></div>
<h3>Generalizations</h3>
<p>We can think of these equations for <span class="math">\(\boldsymbol{\lambda}\)</span> as a simple <em>linear</em>
system of equations, which we are solving by back-substitution when we use the
backpropagation method. The reason why back-substitution is sufficient for the
linear system (i.e., we don't need a <em>full</em> linear system solver) is that the
dependency graph induced by the <span class="math">\(\alpha\)</span> relation is acyclic. If we had needed a
full linear system solver, the solution would take <span class="math">\(\mathcal{O}(n^3)\)</span> time
instead of linear time, seriously blowing-up our nice runtime!</p>
<p>This connection to linear systems is interesting: It tells us that we can
compute <em>global</em> gradients in cyclic graphs. All we'd need is to run a linear
system solver to stitch together <em>local</em> gradients! That is exactly what the
<a href="https://en.wikipedia.org/wiki/Implicit_function_theorem">implicit function theorem</a>
says!</p>
<p>Cyclic constraints add some expressive powerful to our "constraint language," and
it's interesting that we can still efficiently compute gradients in this
setting. An example of what a general type of cyclic constraint looks like is</p>
<div class="math">$$
\begin{align*}
& \underset{\boldsymbol{x}}{\text{argmax}}\, z_n \\
& \text{s.t.}\quad g(\boldsymbol{z}) = \boldsymbol{0} \\
& \text{and}\quad \boldsymbol{z}_{1:d} = \boldsymbol{x}
\end{align*}
$$</div>
<p>where <span class="math">\(g\)</span> can be any smooth multivariate function of the intermediate variables!
Of course, allowing cyclic constraints comes at the cost of a more-difficult
analogue of "the forward pass" to satisfy the <span class="math">\(\boldsymbol{z}\)</span> equations (if we
want to keep it a block-coordinate step). The <span class="math">\(\boldsymbol{\lambda}\)</span> equations
are now a linear system that requires a linear solver (e.g., Gaussian
elimination).</p>
<p>Example use cases:</p>
<ul>
<li>
<p>Bi-level optimization: Solving an optimization problem with another one inside
it. For example,
<a href="http://timvieira.github.io/blog/post/2016/03/05/gradient-based-hyperparameter-optimization-and-the-implicit-function-theorem/">gradient-based hyperparameter optimization</a>
in machine learning. The implicit function theorem manages to get gradients of
hyperparameters without needing to store any of the intermediate states of the
optimization algorithm used in the inner optimization! This is a <em>huge</em> memory
saver since direct backprop on the inner gradient descent algorithm would
require caching all intermediate states. Yikes!</p>
</li>
<li>
<p>Cyclic constraints are useful in many graph algorithms. For example, computing
gradients of edge weights in a general finite-state machine or, similarly,
computing the value function in a Markov decision process.</p>
</li>
</ul>
<h3>Other methods for optimization?</h3>
<p>The connection to Lagrangians brings tons of algorithms for constrained
optimization into the mix! We can imagine using more general algorithms for
optimizing our function and other ways of enforcing the constraints. We see
immediately that we could run optimization with adjoints set to values other
than those that backprop would set them to (i.e., we can optimize them like we'd
do in other algorithms for optimizing general Lagrangians).</p>
<h2>Summary</h2>
<p>Backprop does not directly fall out of the rules for differentiation that you
learned in calculus (e.g., the chain rule).</p>
<ul>
<li>This is because it operates on a more general family of functions: <em>programs</em>
which have <em>intermediate variables</em>. Supporting intermediate variables is
crucial for implementing both functions and their gradients efficiently.</li>
</ul>
<p>I described how we could use something we did learn from calculus 101, the
method of Lagrange multipliers, to support optimization with intermediate
variables.</p>
<ul>
<li>
<p>It turned out that backprop is a <em>particular instantiation</em> of the method of
Lagrange multipliers, involving block-coordinate steps for solving for the
intermediates and multipliers.</p>
</li>
<li>
<p>I also described a neat generalization to support <em>cyclic</em> programs and I
hinted at ideas for doing optimization a little differently, deviating from
the de facto block-coordinate strategy.</p>
</li>
</ul>
<!--
<center>
![Levels of enlightenment](/blog/images/backprop-brain-meme.png)
</center>
-->
<h2>Further reading</h2>
<p>After working out the connection between backprop and the method of Lagrange
multipliers, I discovered following paper, which beat me to it. I don't think my
version is too redundant.</p>
<blockquote>
<p>Yann LeCun. (1988)
<a href="http://yann.lecun.com/exdb/publis/pdf/lecun-88.pdf">A Theoretical Framework from Back-Propagation</a>.</p>
</blockquote>
<p>Ben Recht has a great blog post that uses the implicit function theorem to
<em>derive</em> the method of Lagrange multipliers. He also touches on the connection
to backpropagation.</p>
<blockquote>
<p>Ben Recht. (2016)
<a href="http://www.argmin.net/2016/05/31/mechanics-of-lagrangians/">Mechanics of Lagrangians</a>.</p>
</blockquote>
<p>Tom Goldstein's group took the Lagrangian view of backprop and used it to design
an ADMM approach for optimizing neural nets. The ADMM approach
can run massively in parallel and can leverage highly optimized solvers for
subproblems. This work nicely demonstrates that understanding automatic
differentiation—in the broader sense that I described in this
post—facilitates the development of novel optimization algorithms. <!--
ADMM is based on a cool reformulation trick, which takes a *big* circuit and
breaks it up into several *small* circuits (subproblems), which are *decoupled*
from the big problem because each subproblem gets to freely tune its own *local*
version of the variables. There is, of course, a global equality constraint on
the decoupled variables so that we get a correct solution. The global equality
constraints iteratively bring the subproblems into agreement.--></p>
<blockquote>
<p>Gavin Taylor, Ryan Burmeister, Zheng Xu, Bharat Singh, Ankit Patel, Tom Goldstein. (2018)
<a href="https://arxiv.org/abs/1605.02026">Training Neural Networks Without Gradients: A Scalable ADMM Approach</a>.</p>
</blockquote>
<p>The backpropagation algorithm can be cleanly generalized from values to
functionals!</p>
<blockquote>
<p>Alexander Grubb and J. Andrew Bagnell. (2010)
<a href="https://t.co/5OW5xBT4Y1">Boosted Backpropagation Learning for Training Deep Modular Networks</a>.</p>
</blockquote>
<h2>Code</h2>
<p>I have coded up and tested the Lagrangian perspective on automatic
differentiation that I presented in this article. The code is available in this
<a href="https://gist.github.com/timvieira/8addcb81dd622b0108e0e7e06af74185">gist</a>.</p>
<script src="https://gist.github.com/timvieira/8addcb81dd622b0108e0e7e06af74185.js"></script>
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</script>Estimating means in a finite universe2017-07-03T00:00:00-04:002017-07-03T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2017-07-03:/blog/post/2017/07/03/estimating-means-in-a-finite-universe/<style>
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<p><strong>Introduction</strong>: In this post, I'm going to describe some efficient approaches
to estimating the mean of a random variable that takes on only finitely many
values. Despite the ubiquity of Monte Carlo estimation, it is really inefficient
for finite domains. I'll describe some lesser-known algorithms based on sampling
without replacement that can be adapted to estimating means.</p>
<p><strong>Setup</strong>: Suppose we want to estimate an expectation of a derministic function
<span class="math">\(f\)</span> over a (large) finite universe of <span class="math">\(n\)</span> elements where each element <span class="math">\(i\)</span> has
probability <span class="math">\(p_i\)</span>:</p>
<div class="math">$$
\mu \overset{\tiny{\text{def}}}{=} \sum_{i=1}^n p_i f(i)
$$</div>
<p>However, <span class="math">\(f\)</span> is too expensive to evaluate <span class="math">\(n\)</span> times. So let's say that we have
<span class="math">\(m \le n\)</span> evaluations to form our estimate. (Obviously, if we're happy
evaluating <span class="math">\(f\)</span> a total of <span class="math">\(n\)</span> times, then we should just compute <span class="math">\(\mu\)</span> exactly
with the definition above.)</p>
<!--
**Why I'm writing this post**: Monte Carlo is often used in designing algorithms
as a means to cheaply approximate intermediate expectations, think of stochastic
gradient descent as a prime example. However, in many cases, we have a *finite*
universe, i.e., we *could* enumerate all elements, but it's just inefficient to
do so. In other words, sampling is merely a choice made by the algorithm
designer, not a fundamental property of the environment, as it is typically in
statistics. What can we do to improve estimation in this special setting? I
won't get into bigger questions of how to design these algorithms, instead I'll
focus on this specific type of estimation problem.
-->
<p><strong>Monte Carlo:</strong> The most well-known approach to this type of problem is Monte
Carlo (MC) estimation: sample <span class="math">\(x^{(1)}, \ldots, x^{(m)}
\overset{\tiny\text{i.i.d.}}{\sim} p\)</span>, return <span class="math">\(\widehat{\mu}_{\text{MC}} =
\frac{1}{m} \sum_{i = 1}^m f(x^{(i)})\)</span>. <em>Remarks</em>: (1) Monte Carlo can be very
inefficient because it resamples high-probability items over and over again. (2)
We can improve efficiency—measured in <span class="math">\(f\)</span> evaluations—somewhat by
caching past evaluations of <span class="math">\(f\)</span>. However, this introduces a serious <em>runtime</em>
inefficiency and requires modifying the method to account for the fact that <span class="math">\(m\)</span>
is not fixed ahead of time. (3) Even in our simple setting, MC never reaches
<em>zero</em> error; it only converges in an <span class="math">\(\epsilon\)</span>-<span class="math">\(\delta\)</span> sense.</p>
<!--
Remarks
- We saw a similar problem where we kept sampling the same individuals over and
over again in my
[sqrt-biased sampling post](http://timvieira.github.io/blog/post/2016/06/28/sqrt-biased-sampling/).
-->
<p><strong>Sampling without replacement:</strong> We can get around the problem of resampling
the same elements multiple times by sampling <span class="math">\(m\)</span> distinct elements. This is
called a sampling <em>without replacement</em> (SWOR) scheme. Note that there is no
unique sampling without replacement scheme; although, there does seem to be a
<em>de facto</em> method (more on that later). There are lots of ways to do sampling
without replacement, e.g., any point process over the universe will do as long
as we can control the size.</p>
<p><strong>An alternative formulation:</strong> We can also formulate our estimation problem as
seeking a sparse, unbiased approximation to a vector <span class="math">\(\boldsymbol{x} \in \mathbb{R}_{>0}^n\)</span>. We want
our approximation, <span class="math">\(\boldsymbol{s}\)</span> to satisfy <span class="math">\(\mathbb{E}[\boldsymbol{s}] =
\boldsymbol{x}\)</span> and while <span class="math">\(|| \boldsymbol{s} ||_0 \le m\)</span>. This will suffice for
estimating <span class="math">\(\mu\)</span> (above) when <span class="math">\(\boldsymbol{x}=\boldsymbol{p}\)</span>, the vector of
probabillties, because <span class="math">\(\mathbb{E}[\boldsymbol{s}^\top\! \boldsymbol{f}] =
\mathbb{E}[\boldsymbol{s}]^\top\! \boldsymbol{f} = \boldsymbol{p}^\top\!
\boldsymbol{f} = \mu\)</span> where <span class="math">\(\boldsymbol{f}\)</span> is a vector of all <span class="math">\(n\)</span> values of
the function <span class="math">\(f\)</span>. Obviously, we don't need to evaluate <span class="math">\(f\)</span> in places where
<span class="math">\(\boldsymbol{s}\)</span> is zero so it works for our budgeted estimation task. Of
course, unbiased estimation of all probabillties is not <em>necessary</em> for unbiased
estimation of <span class="math">\(\mu\)</span> alone. However, this characterization is a good model for
when we have zero knowledge of <span class="math">\(f\)</span>. Additionally, this formulation might be of
independent interest, since a sparse, unbiased representation of a vector might
be useful in some applications (e.g., replacing a dense vector with a sparse
vector can lead to more efficient computations).</p>
<p><strong>Priority sampling</strong>: Priority sampling (Duffield et al., 2005;
<a href="http://nickduffield.net/download/papers/priority.pdf">Duffield et al., 2007</a>)
is a remarkably simple algorithm, which is essentially optimal for our task, if
we assume no prior knowledge about <span class="math">\(f\)</span>. Here is pseudocode for priority sampling
(PS), based on the <em>alternative formulation</em>.</p>
<div class="math">$$
\begin{align*}
&\textbf{procedure } \textrm{PrioritySample} \\
&\textbf{inputs: } \text{vector } \boldsymbol{x} \in \mathbb{R}_{>0}^n, \text{budget } m \in \{1, \ldots, n\}\\
&\textbf{output: } \text{sparse and unbiased representation of $\boldsymbol{x}$} \\
&\quad u_i, \ldots, u_n \overset{\tiny\text{i.i.d.}} \sim \textrm{Uniform}(0,1] \\
&\quad k_i \leftarrow u_i/x_i \text{ for each $i$} \quad\color{grey}{\text{# random sort key }} \\
&\quad S \leftarrow \{ \text{$m$-smallest elements according to $k_i$} \} \\
&\quad \tau \leftarrow (m+1)^{\text{th}}\text{ smallest }k_i \\
&\quad s_i \gets \begin{cases}
\max\left( x_i, 1/\tau \right) & \text{ if } i \in S \\
0 & \text{ otherwise}
\end{cases} \\
&\quad \textbf{return }\boldsymbol{s}
\end{align*}
$$</div>
<p><span class="math">\(\textrm{PrioritySample}\)</span> can be applied to obtain a sparse and unbiased
representation of any vector in <span class="math">\(\mathbb{R}^n\)</span>. We make use of such a
representation for our original problem of budgeted mean estimation (<span class="math">\(\mu\)</span>) as
follows:</p>
<div class="math">$$
\begin{align*}
& \boldsymbol{s} \gets \textrm{PrioritySample}(\boldsymbol{p}, m) \\
& \widehat{\mu}_{\text{PS}} = \sum_{i \in S} s_i \!\cdot\! f(i)
\end{align*}
$$</div>
<p>Explanation: The definition of <span class="math">\(s_i\)</span> might look a little mysterious. In the <span class="math">\((i
\in S)\)</span> case, it comes from <span class="math">\(s_i = \frac{p_i}{p(i \in S | \tau)} =
\frac{p_i}{\min(1, x_i \cdot \tau)} = \max(x_i,\ 1/\tau)\)</span>. The factor <span class="math">\(p(i \in S
| \tau)\)</span> is an importance-weighting correction that comes from the
<a href="https://en.wikipedia.org/wiki/Horvitz%E2%80%93Thompson_estimator">Horvitz-Thompson estimator</a>
(modified slightly from its usual presentation to estimate means),
<span class="math">\(\sum_{i=1}^n \frac{p_i}{q_i} \cdot f(i) \cdot \boldsymbol{1}[ i \in S]\)</span>, where
<span class="math">\(S\)</span> is sampled according to some process with inclusion probabilities <span class="math">\(q_i = p(i
\in S)\)</span>. In the case of priority sampling, we have an auxiliary variable for
<span class="math">\(\tau\)</span> that makes computing <span class="math">\(q_i\)</span> easy. Thus, for priority sampling, we can use
<span class="math">\(q_i = p(i \in S | \tau)\)</span>. This auxillary variable adds a tiny bit extra noise
in our estimator, which is tantamount to one extra sample.</p>
<p><button class="toggle-button" onclick="toggle('#ps-unbiased');">Show proof of
unbiasedness</button> <div id="ps-unbiased" class="derivation"
style="display:none;"> <strong>Proof of unbiasedness</strong>. The following proof is a
little different from that in the priority sampling papers. I think it's more
straightforward. More importantly, it shows how we can extend the method to
sample from slightly different without-replacement distributions (as long as we
can compute <span class="math">\(q_i(\tau) = \mathrm{Pr}(i \in S \mid \tau) = \mathrm{Pr}(k_i \le \tau)\)</span>).</p>
<div class="math">$$
\begin{eqnarray}
\mathbb{E}\left[ \widehat{\mu}_{\text{PS}} \right]
&=& \mathbb{E}_{\tau, k_1, \ldots k_n}\! \left[ \sum_{i=1}^n \frac{p_i}{q_i(\tau)} \cdot f(i) \cdot \boldsymbol{1}[ k_i \le \tau] \right] \\
&=& \mathbb{E}_{\tau}\! \left[ \sum_{i=1}^n \frac{p_i}{q_i(\tau)} \cdot f(i) \cdot \mathbb{E}_{k_i | \tau}\!\Big[ \boldsymbol{1}[ k_i \le \tau ] \Big] \right] \\
&=& \mathbb{E}_{\tau}\! \left[
\sum_{i=1}^n \frac{p_i}{q_i(\tau)} \cdot f(i) \cdot
\mathrm{Pr}( k_i \le \tau )
\right] \\
&=& \mathbb{E}_{\tau}\! \left[
\sum_{i=1}^n \frac{p_i}{q_i(\tau)} \cdot f(i) \cdot
q_i(\tau)
\right] \\
&=& \mathbb{E}_{\tau}\! \left[
\sum_{i=1}^n p_i \cdot f(i)
\right] \\
&=& \sum_{i=1}^n p_i \cdot f(i) \\
&=& \mu
\end{eqnarray}
$$</div>
<p>
</div></p>
<p><strong>Remarks</strong>:</p>
<ul>
<li>
<p>Priority sampling satisfies our task criteria: it is both unbiased and sparse
(i.e., under the evaluation budget).</p>
</li>
<li>
<p>Priority sampling can be straighforwardly generalized to support streaming
<span class="math">\(x_i\)</span>, since the keys and threshold can be computed as we run, which means it
can be stopped at any time, in principle.</p>
</li>
<li>
<p>Priority sampling was designed for estimating subset sums, i.e., estimating
<span class="math">\(\sum_{i \in I} x_i\)</span> for some <span class="math">\(I \subseteq \{1,\ldots,n\}\)</span>. In this setting,
the set of sampled items <span class="math">\(S\)</span> is chosen to be "representative" of the
population, albeit much smaller. In the subset sum setting, priority sampling
has been shown to have near-optimal variance
<a href="https://www.cs.rutgers.edu/~szegedy/PUBLICATIONS/full1.pdf">(Szegedy, 2005)</a>.
Specifically, priority sampling with <span class="math">\(m\)</span> samples is no worse than the best
possible <span class="math">\((m-1)\)</span>-sparse estimator in terms of variance. Of course,
if we have some knowledge about <span class="math">\(f\)</span>, we may be able to beat
PS. <!-- We can relate subset sums to estimating <span class="math">\(\mu\)</span> by interpreting
<span class="math">\(\boldsymbol{x} = \alpha\!\cdot\! \boldsymbol{p}\)</span> for some <span class="math">\(\alpha\)</span>, scaling
<span class="math">\(f\)</span> appropriately by <span class="math">\(\alpha\)</span>, and encoding the subset via indicators in
<span class="math">\(f\)</span>'s dimensions. -->
<!-- (e.g.,. via
<a href="http://timvieira.github.io/blog/post/2016/05/28/the-optimal-proposal-distribution-is-not-p/">importance sampling</a>
or by modifying PS to sample proportional to <span class="math">\(x_i = p_i \!\cdot\! |f_i|\)</span> (as
well as other straightforward modifications), but presumably with a surrogate
for <span class="math">\(f_i\)</span> because we don't want to evaluate it). --></p>
</li>
<li>
<p>Components of the estimate <span class="math">\(\boldsymbol{s}\)</span> are uncorrelated, i.e.,
<span class="math">\(\textrm{Cov}[s_i, s_j] = 0\)</span> for <span class="math">\(i \ne j\)</span> and <span class="math">\(m \ge 2\)</span>. This is surprising
since <span class="math">\(s_i\)</span> and <span class="math">\(s_j\)</span> are related via the threshold <span class="math">\(\tau\)</span>.</p>
</li>
<li>
<p>If we instead sample <span class="math">\(u_1, \ldots, u_n \overset{\text{i.i.d.}}{\sim}
-\textrm{Exponential}(1)\)</span>, then <span class="math">\(S\)</span> will be sampled according to the <em>de facto</em>
sampling without replacement scheme (e.g., <code>numpy.random.sample(..., replace=False)</code>),
known as probability proportional to size without replacement (PPSWOR).
To we can then adjust our estimator
<div class="math">$$
\widehat{\mu}_{\text{PPSWOR}} = \sum_{i \in S} \frac{p_i}{q_i} f(i)
$$</div>
where <span class="math">\(q_i = p(i \in S|\tau) = p(k_i > \tau) = 1-\exp(-x_i \!\cdot\!
\tau)\)</span>. This estimator performs about as well as priority sampling. It
inherits my proof of unbiasedness (above).</p>
</li>
<li>
<p><span class="math">\(\tau\)</span> is an auxiliary variable that is introduced to break complex
dependencies between keys. Computing <span class="math">\(\tau\)</span>'s distribution is complicated
because it is an order statistic of non-identically distributed random
variates; this means we can't rely on symmetry to make summing over
permutations efficient.</p>
</li>
</ul>
<!--
- The one downside of this method is that sampling seems to require looking at
all $n$ items.
-->
<h2>Experiments</h2>
<p>You can get the Jupyter notebook for replicating this experiment
<a href="https://github.com/timvieira/blog/blob/master/content/notebook/Priority%20Sampling.ipynb">here</a>.
So download the notebook and play with it!</p>
<p>The improvement of priority sampling (PS) over Monte Carlo (MC) is pretty
nice. I've also included PPSWOR, which seems pretty indistinguishable from PS so
I won't really bother to discuss it. Check out the results!</p>
<p><center>
<img alt="Priority sampling vs. Monte Carlo" src="http://timvieira.github.io/blog/images/ps-mc.png">
</center></p>
<p>The shaded region indicates the 10% and 90% percentiles over 20,000
replications, which gives a sense of the variability of each estimator. The
x-axis is the sampling budget, <span class="math">\(m \le n\)</span>.</p>
<p>The plot shows a small example with <span class="math">\(n=50\)</span>. We see that PS's variability
actually goes to zero, unlike Monte Carlo, which is still pretty inaccurate even
at <span class="math">\(m=n\)</span>. (Note that MC's x-axis measures raw evaluations, not distinct ones.)</p>
<p><strong>Further reading:</strong> If you liked this post, you might like my other posts
tagged with <a href="http://timvieira.github.io/blog/tag/sampling.html">sampling</a> and
<a href="http://timvieira.github.io/blog/tag/reservoir-sampling.html">reservoir sampling</a>.</p>
<ul>
<li>
<p>Edith Cohen, "The Magic of Random Sampling"
(<a href="http://www.cohenwang.com/edith/Talks/MagicSampling201611.pdf">slides</a>,
<a href="https://www.youtube.com/watch?v=jp83HyDs8fs">talk</a>)</p>
</li>
<li>
<p><a href="http://nickduffield.net/download/papers/priority.pdf">Duffield et al., (2007)</a>
has plenty good stuff that I didn't cover.</p>
</li>
<li>
<p>Alex Smola's <a href="http://blog.smola.org/post/1078486350/priority-sampling">post</a></p>
</li>
<li>
<p>Suresh Venkatasubramanian's
<a href="http://blog.geomblog.org/2005/10/priority-sampling.html">post</a></p>
</li>
</ul>
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</script>How to test gradient implementations2017-04-21T00:00:00-04:002017-04-21T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2017-04-21:/blog/post/2017/04/21/how-to-test-gradient-implementations/<!--
**Who should read this?** Nowadays, you're probably just using automatic
differentiation to compute the gradient of whatever function you're using. If
that's you and you trust your software package wholeheartedly, then you probably
don't need to read this. If you're rolling your own module/Op to put in to an
auto-diff library, then you should read this. I find that none of the available
libraries are any good for differentiating dynamic programs, so I still find
this stuff useful. You'll probably have to write gradient code without the aid
of an autodiff library someday... So either way, knowing this stuff is good for
you. To answer the question, everyone.
-->
<p><strong>Setup</strong>: Suppose we have a function, <span class="math">\(f: \mathbb{R}^n \rightarrow \mathbb{R}\)</span>,
and we want to test code that computes <span class="math">\(\nabla f\)</span>. (Note that these techniques
also apply when <span class="math">\(f\)</span> has multivariate output.)</p>
<h2>Finite-difference approximation</h2>
<p>The main way that people test gradient computation is by comparing it against a
finite-difference (FD) approximation to the gradient:</p>
<div class="math">$$
\boldsymbol{d}^\top\! \nabla f(\boldsymbol{x}) \approx \frac{1}{2 \varepsilon}(f(\boldsymbol{x} + \varepsilon \cdot \boldsymbol{d}) - f(\boldsymbol{x} - \varepsilon \cdot \boldsymbol{d}))
$$</div>
<p>
<br/>
where <span class="math">\(\boldsymbol{d} \in \mathbb{R}^n\)</span> is an arbitrary "direction" in parameter
space. We will look at many directions when we test. Generally, people take the
<span class="math">\(n\)</span> elementary vectors as the directions, but random directions are just as good
(and you can catch bugs in all dimensions with less than <span class="math">\(n\)</span> of them).</p>
<p><strong>Always use the two-sided difference formula</strong>. There is a version which
doesn't add <em>and</em> subtract, just does one or the other. Do not use it ever.</p>
<p><strong>Make sure you test multiple inputs</strong> (values of <span class="math">\(\boldsymbol{x}\)</span>) or any thing
else the function depends on (e.g., the minibatch).</p>
<p><strong>What directions to use</strong>: When debugging, I tend to use elementary directions
because they tell me something about which dimensions that are wrong... this
doesn't always help though. The random directions are best when you want the
test cases to run really quickly. In that case, you can switch to check a few
random directions using a
<a href="https://github.com/timvieira/arsenal/blob/master/arsenal/math/util.py">spherical</a>
distribution—do <em>not</em> sample them from a multivariate uniform!</p>
<p><strong>Always test your implementation of <span class="math">\(f\)</span>!</strong> It's very easy to <em>correctly</em>
compute the gradient of the <em>wrong</em> function. The FD approximation is a
"self-consistency" test, it does not validate <span class="math">\(f\)</span> only the relationship
between <span class="math">\(f\)</span> and <span class="math">\(\nabla\! f\)</span>.</p>
<p>Obviously, how you test <span class="math">\(f\)</span> depends strongly on what it's supposed to compute.</p>
<ul>
<li>Example: For a conditional random field (CRF), you can also test that your
implementation of a dynamic program for computing <span class="math">\(\log Z_\theta(x)\)</span> is
correctly by comparing against brute-force enumeration of <span class="math">\(\mathcal{Y}(x)\)</span> on
small examples.</li>
</ul>
<p>Similarly, you can directly test the gradient code if you know a different way
to compute it.</p>
<ul>
<li>Example: In a CRF, we know that the <span class="math">\(\nabla \log Z_\theta(x)\)</span> is a feature
expectation, which you can also test against a brute-force enumeration on
small examples.</li>
</ul>
<h3>Why not just use the FD approximation as your gradient?</h3>
<p>For low-dimensional functions, you can straight-up use the finite-difference
approximation instead of rolling code to compute the gradient. (Take <span class="math">\(n\)</span>
axis-aligned unit vectors for <span class="math">\(\boldsymbol{d}\)</span>.) The FD approximation is very
accurate. Of course, specialized code is probably a little more accurate, but
that's not <em>really</em> why we bother to do it! The reason why we write specialized
gradient code is <em>not</em> to improve numerical accuracy, it's to improve
<em>efficiency</em>. As I've
<a href="http://timvieira.github.io/blog/post/2016/09/25/evaluating-fx-is-as-fast-as-fx/">ranted</a>
before, automatic differentiation techniques guarantee that evaluating <span class="math">\(\nabla
f(x)\)</span> gradient should be as efficient as computing <span class="math">\(f(x)\)</span> (with the caveat that
<em>space</em> complexity may increase substantially - i.e., space-time tradeoffs
exists). FD is <span class="math">\(\mathcal{O}(n \cdot \textrm{runtime } f(x))\)</span>, where as autodiff
is <span class="math">\(\mathcal{O}(\textrm{runtime } f(x))\)</span>.</p>
<h2>How to compare vectors</h2>
<p><strong>Absolute difference is the devil.</strong> You should never compare vectors in
absolute difference (this is Lecture 1 of any numerical methods course). In this
case, the problem is that gradients depend strongly on the scale of <span class="math">\(f\)</span>. If <span class="math">\(f\)</span>
takes tiny values then it's easy for differences to be lower than a tiny
threshold.</p>
<p>Most people use <strong>relative error</strong> <span class="math">\(= \frac{|\textbf{want} -
\textbf{got}|}{|\textbf{want}|}\)</span>, to get a scale-free error measure, but
unfortunately relative error chokes when <span class="math">\(\textbf{want}\)</span> is zero.</p>
<p>I compute several error measures with a script that you can import from my
github
<a href="https://github.com/timvieira/arsenal/blob/master/arsenal/math/checkgrad.py">arsenal.math.checkgrad.{fdcheck}</a>.</p>
<p>I use two metrics to test gradients:</p>
<ol>
<li>
<p>Relative error (skipping zeros): If relative error hits a zero, I skip
it. I'll rely on the other measure.</p>
</li>
<li>
<p>Pearson correlation: Checks the <em>direction</em> of the gradient, but allows a
scale and shift transformation. This measure doesn't have trouble with zeros,
but allows scale and shift problems to pass by. <em>Make sure you fix those
errors!</em> (e.g. In the CRF example, you might have forgotten to divide by
<span class="math">\(Z(x)\)</span>, which not really a constant... I've made this exact mistake a few
times.)</p>
</li>
</ol>
<p>I also look at some diagnostics, which help me debug stuff:</p>
<ul>
<li>
<p>Accuracy of predicting the sign {+,-,0} of each dimension (or dot random product).</p>
</li>
<li>
<p>Absolute error (just as a diagnostic)</p>
</li>
<li>
<p>Scatter plot: When debugging, I like to scatter plot the elements of FD vs. my
implementation.</p>
</li>
</ul>
<p>All these measurements (and the scatter plot) can be computed with
<a href="https://github.com/timvieira/arsenal/blob/master/arsenal/math/compare.py">arsenal.math.compare.{compare}</a>,
which I find super useful when debugging absolutely anything numerical.</p>
<h2>Bonus tests</h2>
<p><strong>Testing modules</strong>: You can test the different modules of your code as well
(assuming you have a composable module-based setup). E.g., I test my DP
algorithm independent of how the features and downstream loss are computed. You
can also test feature and downstream loss modules independent of one
another. Note that autodiff (implicitly) computes Jacobian-vector products
because modules are multivariate in general. We can reduce to the scalar case by
taking a dot product of the outputs with a (fixed) random vector.</p>
<p>Something like this:</p>
<div class="highlight"><pre><span></span><code><span class="n">r</span> <span class="o">=</span> <span class="n">spherical</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="c1"># fixed random vector |output|=|m|</span>
<span class="n">h</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">module</span><span class="o">.</span><span class="n">fprop</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="c1"># scalar function for use in fd</span>
<span class="n">module</span><span class="o">.</span><span class="n">fprop</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c1"># propagate</span>
<span class="n">module</span><span class="o">.</span><span class="n">outputs</span><span class="o">.</span><span class="n">adjoint</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span> <span class="c1"># set output adjoint to r, usually we set adjoint of scalar output=1</span>
<span class="n">module</span><span class="o">.</span><span class="n">bprop</span><span class="p">()</span>
<span class="n">ad</span> <span class="o">=</span> <span class="n">module</span><span class="o">.</span><span class="n">input</span><span class="o">.</span><span class="n">adjoint</span> <span class="c1"># grab the gradient</span>
<span class="n">fd</span> <span class="o">=</span> <span class="n">fdgrad</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="n">compare</span><span class="p">(</span><span class="n">fd</span><span class="p">,</span> <span class="n">ad</span><span class="p">)</span>
</code></pre></div>
<p><strong>Integration tests</strong>: Test that running a gradient-based optimization algorithm
is successful with your gradient implementation. Use smaller versions of your
problem if possible. A related test for machine learning applications is to make
sure that your model and learning procedure can (over)fit small datasets.</p>
<p><strong>Test that batch = minibatch</strong> (if applicable). It's very easy to get this bit
wrong. Broadcasting rules (in numpy, for example) make it easy to hide matrix
conformability mishaps. So make sure you get the same results as manual
minibatching (Of course, you should only do minibatching if are get a speed-up
from vectorization or parallelism. You should probably test that it's actually
faster.)</p>
<!--
Other common sources of bugs
* Really look over your test cases. I often find that my errors are actually in
the test case themselves because either (1) I wrote it really quickly with
less care than the difficult function/gradient, or (2) there is a gap between
"what I want it to do" and "what I told it to do".
* Random search in the space of programs can result in overfitting! This is a
general problem with test-driven development that always applies. If you are
hamfistedly twiddling bits of your code without thinking about why things
work, you can trick almost any test.
-->
<p><strong>Further reading</strong>:</p>
<ul>
<li>
<p>I've written about gradient approximations before, you might like these
articles:
<a href="http://timvieira.github.io/blog/post/2014/02/10/gradient-vector-product/">Gradient-vector products</a>,
<a href="http://timvieira.github.io/blog/post/2014/08/07/complex-step-derivative/">Complex-step method</a>.</p>
</li>
<li>
<p>The foundations of backprop:
<a href="http://timvieira.github.io/blog/post/2017/08/18/backprop-is-not-just-the-chain-rule/">Backprop is not just the chain rule</a>.</p>
</li>
<li>
<p><a href="http://timvieira.github.io/blog/post/2016/09/25/evaluating-fx-is-as-fast-as-fx/">I strongly recommend</a>
learning how automatic differentiation works, I learned it from
<a href="https://people.cs.umass.edu/~domke/courses/sml2011/08autodiff_nnets.pdf">Justin Domke's course notes</a>.</p>
</li>
<li>
<p><a href="https://justindomke.wordpress.com/2017/04/22/you-deserve-better-than-two-sided-finite-differences/">Justin Domke post</a>:
Explains why we need bespoke finite-difference stencils (i.e., more than
two-sided differences) to prevent numerical demons from destroying our results!</p>
</li>
</ul>
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</script>Counterfactual reasoning and learning from logged data2016-12-19T00:00:00-05:002016-12-19T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2016-12-19:/blog/post/2016/12/19/counterfactual-reasoning-and-learning-from-logged-data/<style> .toggle-button { background-color: #555555; border: none; color: white;
padding: 10px 15px; border-radius: 6px; text-align: center; text-decoration:
none; display: inline-block; font-size: 16px; cursor: pointer; } .derivation {
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margin-bottom: 10px; } </style>
<script>
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function toggle(x) { $(x).toggle(); }
</script>
<p>Counterfactual reasoning is <em>reasoning about data that we did not observe</em>. For
example, reasoning about the expected reward of new policy given data collected
from a older one.</p>
<p>In this post, I'll discuss some basic techniques for learning from logged
data. For the large part, this post is based on things I learned from
<a href="http://www.cs.ucr.edu/~cshelton/papers/docs/icml02.pdf">Peshkin & Shelton (2002)</a>
and <a href="https://arxiv.org/abs/1209.2355">Bottou et al. (2013)</a> (two of my all-time
favorite papers).</p>
<p>After reading, have a look at the
<a href="https://gist.github.com/timvieira/788c2c25c94663c49abada60f2e107e9">Jupyter notebook</a>
accompanying this post!</p>
<p><strong>Setup</strong> (<em>off-line off-policy optimization</em>): We're trying to optimize a
function of the form,</p>
<div class="math">$$
J(\theta) = \underset{p_\theta}{\mathbb{E}} \left[ r(x) \right] = \sum_{x \in \mathcal{X}} p_\theta(x) r(x).
$$</div>
<p><br/> But! We only have a <em>fixed</em> sample of size <span class="math">\(m\)</span> from a data collection
policy <span class="math">\(q\)</span>, <span class="math">\(\{ (r^{(j)}, x^{(j)} ) \}_{j=1}^m \overset{\text{i.i.d.}} \sim q.\)</span></p>
<ul>
<li>
<p>Although, it's not <em>necessarily</em> the case, you can think of <span class="math">\(q = p_{\theta'}\)</span>
for a <em>fixed</em> value <span class="math">\(\theta'.\)</span></p>
</li>
<li>
<p><span class="math">\(\mathcal{X}\)</span> is an arbitrary multivariate space, which permits a mix of
continuous and discrete components, with appropriate densities, <span class="math">\(p_{\theta}\)</span>
and <span class="math">\(q\)</span> defined over it.</p>
</li>
<li>
<p><span class="math">\(r: \mathcal{X} \mapsto \mathbb{R}\)</span> is a black box that outputs a scalar
score.</p>
</li>
<li>
<p>I've used the notation <span class="math">\(r^{(j)}\)</span> instead of <span class="math">\(r(x^{(j)})\)</span> to emphasize that we
can't evaluate <span class="math">\(r\)</span> at <span class="math">\(x\)</span> values other than those in the sample.</p>
</li>
<li>
<p>We'll assume that <span class="math">\(q\)</span> assigns positive probability everywhere, <span class="math">\(q(x) > 0\)</span> for
all <span class="math">\(x \in \mathcal{X}\)</span>. This means is that the data collection process must
be randomized and eventually sample all possible configurations. Later, I
discuss relaxing this assumption.</p>
</li>
</ul>
<p>Each distribution is a product of one or more factors of the following types:
<strong>policy factors</strong> (at least one), which directly depend on <span class="math">\(\theta\)</span>, and
<strong>environment factors</strong> (possibly none), which do not depend directly on
<span class="math">\(\theta\)</span>. Note that environment factors are <em>only</em> accessible via sampling
(i.e., we don't know the <em>value</em> they assign to a sample). For example, a
<em>contextual bandit problem</em>, where <span class="math">\(x\)</span> is a context-action pair, <span class="math">\(x =
(s,a)\)</span>. Here <span class="math">\(q(x) = q(a|s) p(s)\)</span> and <span class="math">\(p_{\theta}(x) = p_{\theta}(a|s)
p(s)\)</span>. Note that <span class="math">\(p_{\theta}\)</span> and <span class="math">\(q\)</span> share the environment factor <span class="math">\(p(s)\)</span>, the
distribution over contexts, and only differ in the action-given-context
factor. For now, assume that we can evaluate all environment factors; later,
I'll discuss how we cleverly work around it.</p>
<!--
(We can
even extend it to a full-blown MDP or POMDP by taking $x$ to be a sequence of
state-action pairs, often called "trajectories".)
-->
<p><strong>The main challenge</strong> of this setting is that we don't have controlled
experiments to learn from because <span class="math">\(q\)</span> is not (completely) in our control. This
manifests itself as high variance ("noise") in estimating <span class="math">\(J(\theta)\)</span>. Consider
the contextual bandit setting, we receive a context <span class="math">\(s\)</span> and execute single
action; we never get to rollback to that precise context and try an alternative
action (to get a paired sample #yolo) because we do not control <span class="math">\(p(s)\)</span>. This is
an important paradigm for many 'real world' problems, e.g., predicting medical
treatments or ad selection.</p>
<!--
This is the crucial difference that makes counterfactual learning
more difficult than (fully) supervised learning.
-->
<p><strong>Estimating <span class="math">\(J(\theta)\)</span></strong> [V1]: We obtain an unbiased estimator of <span class="math">\(J(\theta)\)</span>
with
<a href="http://timvieira.github.io/blog/post/2014/12/21/importance-sampling/">importance sampling</a>,</p>
<div class="math">$$
J(\theta)
\approx \hat{J}_{\!\text{IS}}(\theta)
= \frac{1}{m} \sum_{j=1}^m r^{(j)} \!\cdot\! w^{(j)}_{\theta}
\quad \text{ where } w^{(j)}_{\theta} = \frac{p_{\theta}(x^{(j)}) }{ q(x^{(j)}) }.
$$</div>
<p><br/> This estimator is remarkable: it uses importance sampling as a function
approximator! We have an <em>unbiased</em> estimate of <span class="math">\(J(\theta)\)</span> for any value of
<span class="math">\(\theta\)</span> that we like. <em>The catch</em> is that we have to pick <span class="math">\(\theta\)</span> a priori,
i.e., with no knowledge of the sample.</p>
<!--
We also require that the usual 'support
conditions' for importance sampling conditions ($p_{\theta}(x)>0 \Rightarrow
q(x)>0$ for all $x \in \mathcal{X}$, which is why we made assumption A1.
-->
<p>After we've collected a (large) sample it's possible to optimize
<span class="math">\(\hat{J}_{\!\text{IS}}\)</span> using any optimization algorithm (e.g., L-BFGS). Of
course, we risk overfitting to the sample if we evaluate
<span class="math">\(\hat{J}_{\!\text{IS}}\)</span>. Actually, it's a bit worse: this objective tends to
favor regions of <span class="math">\(\theta\)</span>, which are not well represented in the sample because
the importance sampling estimator has high variance in these regions resulting
from large importance weights (when <span class="math">\(q(x)\)</span> is small and <span class="math">\(p_{\theta}(x)\)</span> is
large, <span class="math">\(w(x)\)</span> is large and consequently so is <span class="math">\(\hat{J}_{\!\text{IS}}\)</span> regardless
of whether <span class="math">\(r(x)\)</span> is high!). Thus, we want some type of "regularization" to keep
the optimizer in regions which are sufficiently well-represented by the sample.</p>
<!--
**Visual example**: We can visualize this phenomena in a simple example. Let $q
= \mathcal{N}(0, \sigma=5)$, $r(x) = 1 \text{ if } x \in [2, 3], 0.2 \text{
otherwise},$ and $p_\theta = \mathcal{N}(\theta, \sigma=1)$. This example is
nice because it let's us plot $x$ and $\theta$ in the same space. This is
generally not the case, because $\mathcal{X}$ may have no connection to $\theta$
space, e.g., $\mathcal{X}$ may be discrete.
**TODO** add plot
-->
<p><strong>Better surrogate</strong> [V2]: There are many ways to improve the variance of the
estimator and <em>confidently</em> obtain improvements to the system. One of my
favorites is Bottou et al.'s lower bound on <span class="math">\(J(\theta)\)</span>, which we get by
clipping importance weights, replace <span class="math">\(w^{(j)}_{\theta}\)</span> with <span class="math">\(\min(R,
w^{(j)}_{\theta})\)</span>.</p>
<p><strong>Confidence intervals</strong> [V3]: We can augment the V2 lower bound with confidence
intervals derived from the empirical Bernstein bound (EBB). We'll require that
<span class="math">\(r\)</span> is bounded and that we know its max/min values. The EBB <em>penalizes</em>
hypotheses (values of <span class="math">\(\theta\)</span>) which have higher sample variance. (Note: a
Hoeffding bound wouldn't change the <em>shape</em> of the objective, but EBB does
thanks to the sample variance penalty. EBB tends to be tighter.). The EBB
introduces an additional "confidence" hyperparameter, <span class="math">\((1-\delta)\)</span>. Bottou et
al. recommend maximizing the lower bound as it provides safer improvements. See
the original paper for the derivation.</p>
<!--
An important benefit of having upper *and* lower is that the bounds tell
us whether or not we should collect more data
-->
<p>Both V2 and V3 are <em>biased</em> (as they are lower bounds), but we can mitigate the
bias by <em>tuning</em> the hyperparameter <span class="math">\(R\)</span> on a heldout sample (we can even tune
<span class="math">\(\delta\)</span>, if desired). Additionally, V2 and V3 are 'valid' when <span class="math">\(q\)</span> has limited
support since they prevent the importance weights from exploding (of course, the
bias can be arbitrarily bad, but probably unavoidable given the
learning-from-only-logged data setup).</p>
<h2>Extensions</h2>
<!--
**Be warned**: This may be considered an idealized setting. Much of the research
in counterfactual and causal reasoning targets (often subtle) deviations from
these assumptions (and some different questions, of course). Some extensions and
discussion appear towards the end of the post.
-->
<p><strong>Unknown environment factors</strong>: Consider the contextual bandit setting
(mentioned above). Here <span class="math">\(p\)</span> and <span class="math">\(q\)</span> share an <em>unknown</em> environment factor: the
distribution of contexts. Luckily, we do not need to know the value of this
factor in order to apply any of our estimators because they are all based on
likelihood <em>ratios</em>, thus the shared unknown factors cancel out! Some specific
examples are given below. Of course, these factors do influence the estimators
because they are crucial in <em>sampling</em>, they just aren't necessary in
<em>evaluation</em>.</p>
<ul>
<li>
<p>In contextual bandit example, <span class="math">\(x\)</span> is a state-action pair, <span class="math">\(w_{\theta}(x) =
\frac{p_{\theta}(x)}{q(x)} = \frac{ p_{\theta}(s,a) }{ q(s,a) } =
\frac{p_{\theta}(a|s) p(s)}{q(a|s) p(s)} = \frac{p_{\theta}(a|s) }{ q(a|s)
}\)</span>.</p>
</li>
<li>
<p>In a Markov decision process, <span class="math">\(x\)</span> is a sequence of state-action pairs,
<span class="math">\(w_{\theta}(x) = \frac{p_{\theta}(x)}{q(x)} = \frac{ p(s_0) \prod_{t=0}^T
p(s_{t+1}|s_t,a_t) p_\theta(a_t|s_t) } { p(s_0) \prod_{t=0}^T
p(s_{t+1}|s_t,a_t) q(a_t|s_t) } = \frac{\prod_{t=0}^T \pi_\theta(a_t|s_t)}
{\prod_{t=0}^T q(a_t|s_t)}.\)</span></p>
</li>
</ul>
<p><strong>Variance reduction</strong>: These estimators can all be improved with variance
reduction techniques. Probably the most effective technique is using
<a href="https://en.wikipedia.org/wiki/Control_variates">control variates</a> (of which
baseline functions are a special case). These are random variables correlated
with <span class="math">\(r(x)\)</span> for which we know their expectations (or at least they are estimated
separately). A great example is how ad clicks depend strongly on time-of-day
(fewer people are online late at night so we get fewer clicks), thus the
time-of-day covariate explains a large part of the variation in <span class="math">\(r(x)\)</span>.</p>
<p><strong>Estimation instead of optimization</strong>: You can use this general setup for
estimation instead of optimization, in which case it's fine to let <span class="math">\(r\)</span> have
real-valued multivariate output. The confidence intervals are probably useful in
that setting too.</p>
<p><strong>Unknown <span class="math">\(q\)</span></strong>: Often <span class="math">\(q\)</span> is an existing complex system, which does not record
its probabilities. It is possible to use regression to estimate <span class="math">\(q\)</span> from the
samples, which is called the <strong>propensity score</strong> (PS). PS attempts to account
for <strong>confounding variables</strong>, which are hidden causes that control variation in
the data. Failing to account for confounding variables may lead to
<a href="https://en.wikipedia.org/wiki/Simpson's_paradox">incorrect conclusions</a>. Unfortunately,
PS results in a biased estimator because we're using a 'ratio of expectations'
(we'll divide by the PS estimate) instead of an 'expectation of ratios'. PS is
only statistically consistent in the (unlikely) event that the density estimate
is correctly specified (i.e., we can eventually get <span class="math">\(q\)</span> correct). In the unknown
<span class="math">\(q\)</span> setting, it's often better to use the <strong>doubly-robust estimator</strong> (DR) which
combines <em>two</em> estimators: a density estimator for <span class="math">\(q\)</span> and a function
approximation for <span class="math">\(r\)</span>. A great explanation for the bandit case is in
<a href="https://arxiv.org/abs/1103.4601">Dudík et al. (2011)</a>. The DR estimator is also
biased, but it has a better bias-variance tradeoff than PS.</p>
<p><strong>What if <span class="math">\(q\)</span> doesn't have support everywhere?</strong> This is an especially important
setting because it is often the case that data collection policies abide by some
<strong>safety regulations</strong>, which prevent known bad configurations. In many
situations, evaluating <span class="math">\(r(x)\)</span> corresponds to executing an action <span class="math">\(x\)</span> in the real
world so terrible outcomes could occur, such as, breaking a system, giving a
patient a bad treatment, or losing money. V1 is ok to use as long as we satisfy
the importance sampling support conditions, which might mean rejecting certain
values for <span class="math">\(\theta\)</span> (might be non-trivial to enforce) and consequently finding a
less-optimal policy. V2 and V3 are ok to use without an explicit constraint, but
additional care may be needed to ensure specific safety constraints are
satisfied by the learned policy.</p>
<p><strong>What if <span class="math">\(q\)</span> is deterministic?</strong> This is related to the point above. This is a
hard problem. Essentially, this trying to learn without any exploration /
experimentation! In general, we need exploration to learn. Randomization isn't
the only way to perform exploration, there are many systematic types of
experimentation.</p>
<ul>
<li>
<p>There are some cases of systematic experimentation that are ok. For example,
enumerating all elements of <span class="math">\(\mathcal{X}\)</span> (almost certainly
infeasible). Another example is a contextual bandit where <span class="math">\(q\)</span> assigns
actions to contexts deterministically via a hash function (this setting is
fine because <span class="math">\(q\)</span> is essentially a uniform distribution over actions, which
is independent of the state). In other special cases, we <em>may</em> be able to
characterize systematic exploration as
<a href="https://en.wikipedia.org/wiki/Stratified_sampling">stratified sampling</a>.</p>
</li>
<li>
<p>A generic solution might be to apply the doubly-robust estimator, which
"smooths out" deterministic components (by pretending they are random) and
accounting for confounds (by explicitly modeling them in the propensity
score).</p>
</li>
</ul>
<p><strong>What if we control data collection (<span class="math">\(q\)</span>)?</strong> This is an interesting
setting. Essentially, it asks "how do we explore/experiment optimally (and
safely)?". In general, this is an open question and depends on many
considerations, such as, how much control, exploration cost (safety constraints)
and prior knowledge (of <span class="math">\(r\)</span> and unknown factors in the environment). I've seen
some papers cleverly design <span class="math">\(q\)</span>. The first that comes to mind is
<a href="https://graphics.stanford.edu/projects/gpspaper/gps_full.pdf">Levine & Koltun (2013)</a>. Another
setting is <em>online</em> contextual bandits, in which algorithms like
<a href="http://jmlr.org/proceedings/papers/v15/beygelzimer11a/beygelzimer11a.pdf">EXP4</a>
and
<a href="http://www.research.rutgers.edu/~lihong/pub/Chapelle12Empirical.pdf">Thompson sampling</a>,
prescribe certain types of exploration and work interactively (i.e., they don't
have a fixed training sample). Lastly, I'll mention that there are many
techniques for variance reduction by importance sampling, which may apply.</p>
<h2>Further reading</h2>
<blockquote>
<p>Léon Bottou, Jonas Peters, Joaquin Quiñonero-Candela, Denis X. Charles, D. Max
Chickering, Elon Portugaly, Dipankar Ray, Patrice Simard, Ed Snelson.
<a href="https://arxiv.org/abs/1209.2355">Counterfactual reasoning in learning systems</a>.
JMLR 2013.</p>
</blockquote>
<p>The source for the majority of this post. It includes many other interesting
ideas and goes more in depth into some of the details.</p>
<blockquote>
<p>Miroslav Dudík, John Langford, Lihong Li.
<a href="https://arxiv.org/abs/1103.4601">Doubly robust policy evaluation and learning</a>.
ICML 2011.</p>
</blockquote>
<p>Discussed in extensions section.</p>
<blockquote>
<p>Philip S. Thomas.
<a href="http://psthomas.com/papers/Thomas2015c.pdf">Safe reinforcement learning</a>.
PhD Thesis 2015.</p>
</blockquote>
<p>Covers confidence intervals for policy evaluation similar to Bottou et al., as
well as learning algorithms for RL with safety guarantees (e.g., so we don't
break the robot).</p>
<blockquote>
<p>Peshkin and Shelton.
<a href="http://www.cs.ucr.edu/~cshelton/papers/docs/icml02.pdf">Learning from scarce experience</a>.
ICML 2002.</p>
</blockquote>
<p>An older RL paper, which covers learning from logged data. This is one of the
earliest papers on learning from logged data that I could find.</p>
<blockquote>
<p>Levine and Koltun.
<a href="https://graphics.stanford.edu/projects/gpspaper/gps_full.pdf">Guided policy search</a>.
ICML 2013.</p>
</blockquote>
<p>Discusses clever choices for <span class="math">\(q\)</span> to better-guide learning in the RL setting.</p>
<blockquote>
<p>Corinna Cortes, Yishay Mansour, Mehryar Mohri.
<a href="https://papers.nips.cc/paper/4156-learning-bounds-for-importance-weighting.pdf">Learning bounds for importance weighting</a>.
NIPS 2010.</p>
</blockquote>
<p>Discusses <em>generalization bounds</em> for the counterfactual objective. Includes an
alternative weighting scheme to keep importance weights from exploding.</p>
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</script>Heaps for incremental computation2016-11-21T00:00:00-05:002016-11-21T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2016-11-21:/blog/post/2016/11/21/heaps-for-incremental-computation/<p>In this post, I'll describe a neat trick for maintaining a summary quantity
(e.g., sum, product, max, log-sum-exp, concatenation, cross-product) under
changes to its inputs. The trick and it's implementation are inspired by the
well-known max-heap datastructure. I'll also describe a really elegant
application to fast sampling under an evolving categorical distribution.</p>
<p><strong>Setup</strong>: Suppose we'd like to efficiently compute a summary quantity under
changes to its <span class="math">\(n\)</span>-dimensional input vector <span class="math">\(\boldsymbol{w}\)</span>. The particular
form of the quantity we're going to compute is <span class="math">\(z = \bigoplus_{i=1}^n w_i\)</span>,
where <span class="math">\(\oplus\)</span> is some associative binary operator with identity element
<span class="math">\(\boldsymbol{0}\)</span>.</p>
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<p><button class="toggle-button" onclick="toggle('#operator-mathy');">more formally...</button>
<div id="operator-mathy" class="derivation" style="display:none"></p>
<ul>
<li>
<p><span class="math">\(\boldsymbol{w} \in \boldsymbol{K}^n\)</span></p>
</li>
<li>
<p><span class="math">\(\oplus: \boldsymbol{K} \times \boldsymbol{K} \mapsto \boldsymbol{K}\)</span>.</p>
</li>
<li>
<p>Associative: <span class="math">\((a \oplus b) \oplus c = a \oplus (b \oplus c)\)</span> for all <span class="math">\(a,b,c
\in \boldsymbol{K}\)</span>.</p>
</li>
<li>
<p>Identity element: <span class="math">\(\boldsymbol{0} \in \boldsymbol{K}\)</span> such that <span class="math">\(k \oplus
\boldsymbol{0} = \boldsymbol{0} \oplus k = k\)</span>, for all <span class="math">\(k \in \boldsymbol{K}\)</span>.</p>
</li>
</ul>
</div>
<p><strong>The trick</strong>: Essentially, the trick boils down to <em>parenthesis placement</em> in
the expression which computes <span class="math">\(z\)</span>. A freedom we assumed via the associative
property.</p>
<p>I'll demonstrate by example with <span class="math">\(n=8\)</span>.</p>
<p>Linear structure: We generally compute something like <span class="math">\(z\)</span> with a simple
loop. This looks like a right-branching binary tree when we think about the
order of operations,</p>
<div class="math">$$
z = (((((((w_1 \oplus w_2) \oplus w_3) \oplus w_4) \oplus w_5) \oplus w_6) \oplus w_7) \oplus w_8).
$$</div>
<p><br/> Heap structure: Here the parentheses form a balanced tree, which looks
much more like a recursive implementation that computes the left and right
halves and <span class="math">\(\oplus\)</span>s the results (divide-and-conquer style),</p>
<div class="math">$$
z = (((w_1 \oplus w_2) \oplus (w_3 \oplus w_4)) \oplus ((w_5 \oplus w_6) \oplus (w_7 \oplus w_8))).
$$</div>
<p><br/>
The benefit of the heap structure is that there are <span class="math">\(\mathcal{O}(\log n)\)</span>
intermediate quantities that depend on any input, whereas the linear structure
has <span class="math">\(\mathcal{O}(n)\)</span>. The intermediate quantities correspond to the values of each of the
parenthesized expressions.</p>
<p>Since fewer intermediate quantities depend on a given input, fewer intermediates
need to be adjusted upon a change to the input. Therefore, we get faster
algorithms for <em>maintaining</em> the output quantity <span class="math">\(z\)</span> as the inputs change.</p>
<p><strong>Heap datastructure</strong> (aka
<a href="https://en.wikipedia.org/wiki/Fenwick_tree">binary index tree or Fenwick tree</a>):
We're going to store the values of the intermediates quantities and inputs in a
heap datastructure, which is a <em>complete</em> binary tree. In our case, the tree has
depth <span class="math">\(1 + \lceil \log_2 n \rceil\)</span>, with the values of <span class="math">\(\boldsymbol{w}\)</span> at it's
leaves (aligned left) and padding with <span class="math">\(\boldsymbol{0}\)</span> for remaining
leaves. Thus, the array's length is <span class="math">\(< 4 n\)</span>.</p>
<p>This structure makes our implementation really nice and efficient because we
don't need pointers to find the parent or children of a node (i.e., no need to
wrap elements into a "node" class like in a general tree data structure). So, we
can pack everything into an array, which means our implementation has great
memory/cache locality and low storage overhead.</p>
<p>Traversing the tree is pretty simple: Let <span class="math">\(d\)</span> be the number of internal nodes,
nodes <span class="math">\(1 \le i \le d\)</span> are internal. For node <span class="math">\(i\)</span>, left child <span class="math">\(\rightarrow {2
\cdot i},\)</span> right child <span class="math">\(\rightarrow {2 \cdot i + 1},\)</span> parent <span class="math">\(\rightarrow
\lfloor i / 2 \rfloor.\)</span> (Note that these operations assume the array's indices
start at <span class="math">\(1\)</span>. We generally fake this by adding a dummy node at position <span class="math">\(0\)</span>,
which makes implementation simpler.)</p>
<p><strong>Initializing the heap</strong>: Here's code that initializes the heap structure we
just described.</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">sumheap</span><span class="p">(</span><span class="n">w</span><span class="p">):</span>
<span class="s2">"Create sumheap from weights `w` in O(n) time."</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">w</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">d</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span> <span class="c1"># number of intermediates</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">d</span><span class="p">)</span> <span class="c1"># intermediates + leaves</span>
<span class="n">S</span><span class="p">[</span><span class="n">d</span><span class="p">:</span><span class="n">d</span><span class="o">+</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">w</span> <span class="c1"># store `w` at leaves.</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">d</span><span class="p">)):</span>
<span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">S</span><span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
<span class="k">return</span> <span class="n">S</span>
</code></pre></div>
<p><strong>Updating <span class="math">\(w_k\)</span></strong> boils down to fixing intermediate sums that (transitively)
depend on <span class="math">\(w_k.\)</span> I won't go into all of the details here, instead I'll give
code (below). I'd like to quickly point out that the term "parents" is not
great for our purposes because they are actually the <em>dependents</em>: when an
input changes the value the parents, grand parents, great grand parents, etc,
become stale and need to be recomputed bottom up (from the leaves). The code
below implements the update method for changing the value of <span class="math">\(w_k\)</span> and runs in
<span class="math">\(\mathcal{O}(\log n)\)</span> time.</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">update</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
<span class="s2">"Update w[k] = v` in time O(log n)."</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">i</span> <span class="o">=</span> <span class="n">d</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="n">k</span>
<span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
<span class="k">while</span> <span class="n">i</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span> <span class="c1"># fix parents in the tree.</span>
<span class="n">i</span> <span class="o">//=</span> <span class="mi">2</span>
<span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">S</span><span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
</code></pre></div>
<h2>Remarks</h2>
<ul>
<li>
<p><strong>Numerical stability</strong>: If the operations are noisy (e.g., floating point
operator), then the heap version may be better behaved. For example, if
operations have an independent, additive noise rate <span class="math">\(\varepsilon\)</span> then noise
of <span class="math">\(z_{\text{heap}}\)</span> is <span class="math">\(\mathcal{O}(\varepsilon \cdot \log n)\)</span>, whereas
<span class="math">\(z_{\text{linear}}\)</span> is <span class="math">\(\mathcal{O}(\varepsilon \cdot n)\)</span>. (Without further
assumptions about the underlying operator, I don't believe you can do better
than that.)</p>
</li>
<li>
<p><strong>Relationship to max-heap</strong>: In the case of a max or min heap, we can avoid
allocating extra space for intermediate quantities because all intermediates
values are equal to exactly one element of <span class="math">\(\boldsymbol{w}\)</span>.</p>
</li>
<li>
<p><strong>Change propagation</strong>: The general idea of <em>adjusting</em> cached intermediate
quantities is a neat idea. In fact, we encounter it each time we type
<code>make</code> at the command line! The general technique goes by many
names—including change propagation, incremental maintenance, and
functional reactive programming—and applies to basically <em>any</em>
side-effect-free computation. However, it's most effective when the
dependency structure of the computation is sparse and requires little
overhead to find and refresh stale values. In our example of computing <span class="math">\(z\)</span>,
these considerations manifest themselves as the heap vs linear structures and
our fast array implementation instead of a generic tree datastructure.</p>
</li>
</ul>
<h2>Generalizations</h2>
<ul>
<li>
<p>No zero? No problem. We don't <em>actually</em> require a zero element. So, it's
fair to augment <span class="math">\(\boldsymbol{K} \cup \{ \textsf{null} \}\)</span> where
<span class="math">\(\textsf{null}\)</span> is distinguished value (i.e., <span class="math">\(\textsf{null} \notin
\boldsymbol{K}\)</span>) that <em>acts</em> just like a zero after we overload <span class="math">\(\oplus\)</span> to
satisfy the definition of a zero (e.g., by adding an if-statement).</p>
</li>
<li>
<p>Generalization to an arbitrary maps instead of fixed vectors is possible with
a "locator" map, which a bijective map from elements to indices in a dense
array.</p>
</li>
<li>
<p>Support for growing and shrinking: We support <strong>growing</strong> by maintaining an
underlying array that is always slightly larger than we need—which
we're <em>already</em> doing in the heap datastructure. Doubling the size of the
underlying array (i.e., rounding up to the next power of two) has the added
benefit of allowing us to grow <span class="math">\(\boldsymbol{w}\)</span> at no asymptotic cost! This
is because the resize operation, which requires an <span class="math">\(\mathcal{O}(n)\)</span> time to
allocate a new array and copying old values, happens so infrequently that
they can be completely amortized. We get of effect of <strong>shrinking</strong> by
replacing the old value with <span class="math">\(\textsf{null}\)</span> (or <span class="math">\(\boldsymbol{0}\)</span>). We can
shrink the underlying array when the fraction of nonzeros dips below
<span class="math">\(25\%\)</span>. This prevents "thrashing" between shrinking and growing.</p>
</li>
</ul>
<h2>Application</h2>
<p><strong>Sampling from an evolving distribution</strong>: Suppose that <span class="math">\(\boldsymbol{w}\)</span>
corresponds to a categorical distributions over <span class="math">\(\{1, \ldots, n\}\)</span> and that we'd
like to sample elements from in proportion to this (unnormalized) distribution.</p>
<p>Other methods like the <a href="http://www.keithschwarz.com/darts-dice-coins/">alias</a> or
inverse CDF methods are efficient after a somewhat costly initialization
step. But! they are not as efficient as the heap sampler when the distribution
is being updated. (I'm not sure about whether variants of alias that support
updates exist.)</p>
<p><center></p>
<table>
<thead>
<tr>
<th>Method</th>
<th>Sample</th>
<th>Update</th>
<th>Init</th>
</tr>
</thead>
<tbody>
<tr>
<td>alias</td>
<td>O(1)</td>
<td>O(n)?</td>
<td>O(n)</td>
</tr>
<tr>
<td>i-CDF</td>
<td>O(log n)</td>
<td>O(n)</td>
<td>O(n)</td>
</tr>
<tr>
<td>heap</td>
<td>O(log n)</td>
<td>O(log n)</td>
<td>O(n)</td>
</tr>
</tbody>
</table>
<p></center></p>
<p>Use cases include</p>
<ul>
<li>
<p><a href="https://en.wikipedia.org/wiki/Gibbs_sampling">Gibbs sampling</a>, where
distributions are constantly modified and sampled from (changes may not be
sparse so YMMV). The heap sampler is used in
<a href="https://arxiv.org/abs/1412.4986">this paper</a>.</p>
</li>
<li>
<p><a href="https://jeremykun.com/2013/11/08/adversarial-bandits-and-the-exp3-algorithm/">EXP3</a>
(<a href="https://en.wikipedia.org/wiki/Multi-armed_bandit">mutli-armed bandit algorithm</a>)
is an excellent example of an algorithm that samples and modifies a single
weight in the distribution.</p>
</li>
<li>
<p><em>Stochastic priority queues</em> where we sample proportional to priority and the
weights on items in the queue may change, elements are possibly removed after
they are sampled (i.e., sampling without replacement), and elements are added.</p>
</li>
</ul>
<p>Again, I won't spell out all of the details of these algorithms. Instead, I'll
just give the code.</p>
<p><strong>Inverse CDF sampling</strong></p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="n">w</span><span class="p">):</span>
<span class="s2">"Ordinary sampling method, O(n) init, O(log n) per sample."</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">w</span><span class="o">.</span><span class="n">cumsum</span><span class="p">()</span> <span class="c1"># build cdf, O(n)</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">uniform</span><span class="p">()</span> <span class="o">*</span> <span class="n">c</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="c1"># random probe, p ~ Uniform(0, z)</span>
<span class="k">return</span> <span class="n">c</span><span class="o">.</span><span class="n">searchsorted</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="c1"># binary search, O(log n)</span>
</code></pre></div>
<p><strong>Heap sampling</strong> is essentially the same, except the cdf is stored as heap,
which is perfect for binary search!</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">hsample</span><span class="p">(</span><span class="n">S</span><span class="p">):</span>
<span class="s2">"Sample from sumheap, O(log n) per sample."</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">//</span><span class="mi">2</span> <span class="c1"># number of internal nodes.</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">uniform</span><span class="p">()</span> <span class="o">*</span> <span class="n">S</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="c1"># random probe, p ~ Uniform(0, z)</span>
<span class="c1"># Use binary search to find the index of the largest CDF (represented as a</span>
<span class="c1"># heap) value that is less than a random probe.</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">while</span> <span class="n">i</span> <span class="o"><</span> <span class="n">d</span><span class="p">:</span>
<span class="c1"># Determine if the value is in the left or right subtree.</span>
<span class="n">i</span> <span class="o">*=</span> <span class="mi">2</span> <span class="c1"># Point at left child</span>
<span class="n">left</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="c1"># Probability mass under left subtree.</span>
<span class="k">if</span> <span class="n">p</span> <span class="o">></span> <span class="n">left</span><span class="p">:</span> <span class="c1"># Value is in right subtree.</span>
<span class="n">p</span> <span class="o">-=</span> <span class="n">left</span> <span class="c1"># Subtract mass from left subtree</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span> <span class="c1"># Point at right child</span>
<span class="k">return</span> <span class="n">i</span> <span class="o">-</span> <span class="n">d</span>
</code></pre></div>
<p><strong>Code</strong>: Complete code and test cases for heap sampling are available in this
<a href="https://gist.github.com/timvieira/da31b56436045a3122f5adf5aafec515">gist</a>. A fast Cython <a href="https://github.com/timvieira/arsenal/blob/master/arsenal/maths/sumheap.pyx">implementation</a> is available in my Python <a href="https://github.com/timvieira/arsenal/">arsenal</a>.</p>
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</script>Reversing a sequence with sublinear space2016-10-01T00:00:00-04:002016-10-01T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2016-10-01:/blog/post/2016/10/01/reversing-a-sequence-with-sublinear-space/<p>Suppose we have a computation which generates sequence of states <span class="math">\(s_1 \ldots
s_n\)</span> according to <span class="math">\(s_{t} = f(s_{t-1})\)</span> where <span class="math">\(s_0\)</span> is given.</p>
<p>We'd like to devise an algorithm, which can reconstruct each point in the
sequence efficiently as we traverse it backwards. You can think of this as
"hitting undo" from the end of the sequence or reversing a singly linked list.</p>
<p>Obviously, we <em>could</em> just record the entire sequence, but if <span class="math">\(n\)</span> is large <em>or</em>
the size of each state is large, this will be infeasible.</p>
<p><strong>Idea 0</strong>: Rerun the forward pass <span class="math">\(n\)</span> times. Runtime <span class="math">\(\mathcal{O}(n^2)\)</span>, space
<span class="math">\(\mathcal{O}(1)\)</span>.</p>
<p><strong>Idea 1</strong>: Suppose we save <span class="math">\(0 < k \le n\)</span> evenly spaced "checkpoint" states.
Clearly, this gives us <span class="math">\(\mathcal{O}(k)\)</span> space, but what does it do to the
runtime? Well, if we are at time <span class="math">\(t\)</span> the we have to "replay" computation from
the last recorded checkpoint to get <span class="math">\(s_t\)</span>, which takes <span class="math">\(O(n/k)\)</span> time. Thus, the
overall runtimes becomes <span class="math">\(O(n^2/k)\)</span>. This runtime is not ideal.</p>
<p><strong>Idea 2</strong>: <em>Idea 1</em> did something kind of silly, within a chunk of size <span class="math">\(n/k\)</span>,
it does each computation multiple times! Suppose we increase the memory
requirement <em>just a little bit</em> to remember the current chunk we're working on,
making it now <span class="math">\(\mathcal{O}(k + n/k)\)</span>. Now, we compute each state at most <span class="math">\(2\)</span>
times: once in the initial sequence and once in the reverse. This implies a
<em>linear</em> runtime. Now, the question: how should we set <span class="math">\(k\)</span> so that we minimize
extra space? Easy! Solve the following little optimization problem:</p>
<div class="math">$$
\underset{k}{\textrm{argmin}}\ k+n/k = \sqrt{n}
$$</div>
<style>
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<p><button onclick="toggle('#derivation-optimal-space')" class="toggle-button">Derivation</button>
<div id="derivation-optimal-space" style="display:none;" class="derivation">
To get the minimum, we solve for <span class="math">\(k\)</span> that sets the derivative to zero.
</p>
<div class="math">$$
\begin{eqnarray}
0 &=& \frac{\partial}{\partial k} \left[ k+n/k \right] \\
&=& 1-n/k^2 \\
n/k^2 &=& 1 \\
k^2 &=& n \\
k &=& \sqrt{n}
\end{eqnarray}
$$</div>
<p>
<br/></p>
<p>Since it's safe to assume that <span class="math">\(n,k \ge 1\)</span> and <span class="math">\(\frac{\partial^2}{\partial k\,
\partial k} = 2 n / k^3 > 0\)</span> this is indeed a minimum. It's also global minimum
because <span class="math">\(k+n/k\)</span> is convex in <span class="math">\(k\)</span> when <span class="math">\(n,k > 0\)</span>.</p>
</div>
<p>That's nuts! We get away with <em>sublinear</em> space <span class="math">\(\mathcal{O}(\sqrt{n})\)</span> and we
only blow up our runtime by a factor of 2. Also, I really love the "introduce a
parameter then optimize it out" trick.</p>
<p><button onclick="toggle('#code-sqrt-space')">Code</button>
<div id="code-sqrt-space" style="display:none;"></p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">sqrt_space</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="n">k</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
<span class="n">memory</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">s0</span>
<span class="n">t</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="n">t</span> <span class="o"><=</span> <span class="n">n</span><span class="p">:</span>
<span class="k">if</span> <span class="n">t</span> <span class="o">%</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">memory</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">=</span> <span class="n">s</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="n">t</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">n</span>
<span class="k">while</span> <span class="n">b</span> <span class="o">>=</span> <span class="n">k</span><span class="p">:</span>
<span class="c1"># last chunk may be shorter than k.</span>
<span class="n">c</span> <span class="o">=</span> <span class="p">((</span><span class="n">n</span> <span class="o">%</span> <span class="n">k</span><span class="p">)</span> <span class="ow">or</span> <span class="n">k</span><span class="p">)</span> <span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="n">n</span> <span class="k">else</span> <span class="n">k</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">step</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">memory</span><span class="p">[</span><span class="n">b</span><span class="o">-</span><span class="n">c</span><span class="p">],</span> <span class="n">c</span><span class="p">)):</span>
<span class="k">yield</span> <span class="n">s</span>
<span class="n">b</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">step</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="s2">"Take `k` steps from state `s`, save path. Cost: O(k) space, O(k) time."</span>
<span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="p">[]</span>
<span class="n">B</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">]</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="n">B</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="k">return</span> <span class="n">B</span>
</code></pre></div>
</div>
<p><strong>Idea 3</strong>: What if we apply "the remember <span class="math">\(k\)</span> states" trick <em>recursively</em>? I'm
going to work this out for <span class="math">\(k=2\)</span> (and then claim that the value of <span class="math">\(k\)</span> doesn't
matter).</p>
<p>Run forward to get the midpoint at <span class="math">\(s_{m}\)</span>, where <span class="math">\(m=b + \lfloor n/2
\rfloor\)</span>. Next, recurse on the left and right chunks <span class="math">\([b,m)\)</span> and <span class="math">\([m,e)\)</span>.
We hit the base case when the width of the interval is
one.</p>
<p>Note that we implicitly store midpoints as we recurse (thanks to the stack
frame). The max depth of the recursion is <span class="math">\(\mathcal{O}(\log n)\)</span>, which gives us
a <span class="math">\(\mathcal{O}(\log n)\)</span> space bound.</p>
<p>We can characterize runtime with the following recurrence relation, <span class="math">\(T(n) = 2
\cdot T(n/2) + \mathcal{O}(n)\)</span>. Since we recognize this as the recurrence for
mergesort, we know that it flattens to <span class="math">\(\mathcal{O}(n \log n)\)</span> time. Also, just
like in the case of sorting, the branching factor doesn't matter so we're happy
with or initial assumption that <span class="math">\(k=2\)</span>.</p>
<p><button onclick="toggle('#code-recursive')">Code</button>
<div id="code-recursive" style="display:none;"></p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">recursive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s0</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
<span class="k">if</span> <span class="n">e</span> <span class="o">-</span> <span class="n">b</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="k">yield</span> <span class="n">s0</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># do O(n/2) work to find the midpoint with O(1) space.</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">s0</span>
<span class="n">d</span> <span class="o">=</span> <span class="p">(</span><span class="n">e</span><span class="o">-</span><span class="n">b</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">recursive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">b</span><span class="o">+</span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
<span class="k">yield</span> <span class="n">s</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">recursive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s0</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">b</span><span class="o">+</span><span class="n">d</span><span class="p">):</span>
<span class="k">yield</span> <span class="n">s</span>
</code></pre></div>
</div>
<h2>Remarks</h2>
<p>The algorithms describe in this post are generic algorithmic tricks, which has
been used in a number of place, including</p>
<ul>
<li>
<p>The classic computer science interview problem of reversing a singly linked list
under a tight budget on <em>additional</em> memory.</p>
</li>
<li>
<p>Backpropagation for computing gradients in sequence models, including HMMs (<a href="https://www.microsoft.com/en-us/research/wp-content/uploads/2000/01/icslp00_logspace.pdf">Zweig & Padmanabhan, 2000</a>)
and RNNs (<a href="https://arxiv.org/abs/1604.06174v2">Chen et al., 2016</a>). I have
sample code that illustrates the basic idea below.</p>
</li>
<li>
<p>Memory-efficient <a href="https://arxiv.org/pdf/cs/0310016v1">omniscient debugging</a>,
which allows a user to inspect program state while moving forward <em>and
backward</em> in time.</p>
</li>
</ul>
<h2>Sample code</h2>
<ul>
<li>
<p><a href="https://gist.github.com/timvieira/d2ac72ec3af7972d2471035011cbf1e2">The basics</a>:
Simple implementation complete with test cases.</p>
</li>
<li>
<p><a href="https://gist.github.com/timvieira/aceb64047aed1b13bf4e4da3b9a4c0ea">Memory-efficient backprop in an RNN</a>:
A simple application with test cases, of course.</p>
</li>
</ul>
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</script>Evaluating ∇f(x) is as fast as f(x)2016-09-25T00:00:00-04:002016-09-25T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2016-09-25:/blog/post/2016/09/25/evaluating-fx-is-as-fast-as-fx/<p>Automatic differentiation ('autodiff' or 'backprop') is great—not just
because it makes it easy to rapidly prototype deep networks with plenty of
doodads and geegaws, but because it means that evaluating the gradient <span class="math">\(\nabla
f(x)\)</span> is as fast of computing <span class="math">\(f(x)\)</span>. In fact, the gradient provably requires at
most a <em>small</em> constant factor more arithmetic operations than the function
itself. Furthermore, autodiff tells us how to derive and implement the gradient
efficiently. This is a fascinating result that is perhaps not emphasized enough
in machine learning.</p>
<p><strong>The gradient should never be asymptotically slower than the function.</strong> In my
recent <a href="/doc/2016-emnlp-vocrf.pdf">EMNLP'16 paper</a>, my coauthors and I found a
line of work on variable-order CRFs
(<a href="https://papers.nips.cc/paper/3815-conditional-random-fields-with-high-order-features-for-sequence-labeling.pdf">Ye+'09</a>;
<a href="http://www.jmlr.org/papers/volume15/cuong14a/cuong14a.pdf">Cuong+'14</a>), which
had an unnecessarily slow and complicated algorithm for computing gradients,
which was asymptotically (and practically) slower than their forward
algorithm. Without breaking a sweat, we derived a simpler and more efficient
gradient algorithm by simply applying backprop to the forward algorithm (and
made some other contributions).</p>
<p><strong>Many algorithms are just backprop.</strong> For example, forward-backward and
inside-outside, are actually just instances of automatic differentiation
(<a href="https://www.cs.jhu.edu/~jason/papers/eisner.spnlp16.pdf">Eisner,'16</a>) (i.e.,
outside is just backprop on inside). This shouldn't be a surprise because these
algorithms are used to compute gradients. Basically, if you know backprop and
the inside algorithm, then you can derive the outside algorithm by applying the
backprop transform manually. I find it easier to understand the outside
algorithm via its connection to backprop, then via
<a href="https://www.cs.jhu.edu/~jason/465/iobasics.pdf">the usual presentation</a>. Note
that inside-outside and forward-backward pre-date backpropagation and have
additional uses beyond computing gradients.</p>
<p><strong>Once you've grokked backprop, the world is your oyster!</strong> You can backprop
through many approximate inference algorithms, e.g.,
<a href="http://www.jmlr.org/proceedings/papers/v15/stoyanov11a/stoyanov11a.pdf">Stoyanov+'11</a>
and many of Justin Domke's papers, to avoid issues I've mentioned
<a href="http://timvieira.github.io/blog/post/2015/02/05/conditional-random-fields-as-deep-learning-models/">before</a>. You
can even backprop through optimization algorithms to get gradients of dev loss wrt
hyperparameters, e.g.,
<a href="http://www.jmlr.org/proceedings/papers/v22/domke12/domke12.pdf">Domke'12</a> and
<a href="https://arxiv.org/abs/1502.03492">Maclaurin+'15</a>.</p>
<p><strong>There's at least one catch!</strong> Although the <em>time</em> complexity of computing the
gradient is as good as the function, the <em>space</em> complexity may be much larger
because the autodiff recipe (at least the default reverse-mode one) requires memoizing
all intermediate quantities (e.g., the quantities you overwrite in a
loop). There are generic methods for balancing the time-space tradeoff in
autodiff, since you can (at least in theory) reconstruct the intermediate
quantities by playing the forward computation again from intermediate
checkpoints (at a cost to runtime, of course). A recent example is
<a href="https://arxiv.org/abs/1606.03401">Gruslys+'16</a>.</p>
<p><strong>A final remark</strong>. Despite the name "automatic" differentiation, there is no
need to rely on software to "automatically" give you gradient routines. Applying
the backprop transformation is generally easy to do manually and sometimes more
efficient than using a library. Many autodiff libraries lack good support for
dynamic computation graph, i.e., when the structure depends on quantities that
vary with the input (e.g., sentence length).</p>
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</script>Fast sigmoid sampling2016-07-04T00:00:00-04:002016-07-04T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2016-07-04:/blog/post/2016/07/04/fast-sigmoid-sampling/<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
</div><div class="inner_cell">
<div class="text_cell_render border-box-sizing rendered_html">
<p>In this notebook, we describe a simple trick for efficiently sampling a Bernoulli random variable $Y$ from a sigmoid-defined distribution, $p(Y = 1) = (1 + \exp(-x))^{-1}$, where $x \in \mathbb{R}$ is the only parameter of the distribution ($x$ is often defined as the dot product of features and weights).</p>
<p>The "slow" method for sampling from a sigmoid,</p>
$$
u \sim \textrm{Uniform}(0,1)
$$$$
Y = \textrm{sigmoid}(x) > u
$$<p>This method is slow because it calls the sigmoid function for every value of $x$. It is slow because $\exp$ is 2-3x slower than basic arithmetic operations.</p>
<p>In this post, I'll describe a simple trick, which is well-suited to vectorized computations (e.g., numpy, matlab). The way it works is by <em>precomputing</em> the expensive stuff (i.e., calls to expensive functions like $\exp$).</p>
$$
\textrm{sigmoid}(x) > u \Leftrightarrow \textrm{logit}(\textrm{sigmoid}(x)) > \textrm{logit}(u) \Leftrightarrow x > \textrm{logit}(u).
$$<p>Some details worth mentioning: (a) <a href="https://en.wikipedia.org/wiki/Logit">logit</a> is the inverse of sigmoid—sometimes it's called <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.expit.html">expit</a> to highlight this connection—and (b) logit is strictly monotonic increasing you can apply it both sides of an inequality and preserve the ordering (there's a plot in the appendix).</p>
<p>The "fast" method derives it's advantage by leveraging the fact that expensive computation can be done independently of the data (i.e., specific values of $x$). The fast method is also interesting as just cute math. In the bonus section of this post, we'll make a connection to the <a href="http://timvieira.github.io/blog/post/2014/07/31/gumbel-max-trick/">Gumbel-max trick</a>.</p>
<p><strong>How fast is it in practice?</strong> Below, we run a quick experiment to test that the method is correct and how fast it is.</p>
</div>
</div>
</div>
<div class="cell border-box-sizing code_cell rendered">
<div class="input">
<div class="prompt input_prompt">In [1]:</div>
<div class="inner_cell">
<div class="input_area">
<div class=" highlight hl-ipython2"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pylab</span> <span class="k">as</span> <span class="nn">pl</span>
<span class="kn">from</span> <span class="nn">numpy.random</span> <span class="kn">import</span> <span class="n">uniform</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">exp</span>
<span class="kn">from</span> <span class="nn">scipy.special</span> <span class="kn">import</span> <span class="n">expit</span> <span class="k">as</span> <span class="n">sigmoid</span><span class="p">,</span> <span class="n">logit</span>
<span class="kn">from</span> <span class="nn">arsenal.timer</span> <span class="kn">import</span> <span class="n">timers</span> <span class="c1"># https://github.com/timvieira/arsenal</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="cell border-box-sizing code_cell rendered">
<div class="input">
<div class="prompt input_prompt">In [2]:</div>
<div class="inner_cell">
<div class="input_area">
<div class=" highlight hl-ipython2"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="n">timers</span><span class="p">()</span>
<span class="c1"># These are the sigmoid parameters we're going to sample from.</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="n">n</span><span class="p">)</span>
<span class="c1"># number of runs to average over.</span>
<span class="n">R</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="c1"># Used for plotting average p(Y=1)</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="c1"># Temporary array for saving on memory allocation, cf. method slow-2.</span>
<span class="n">tmp</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">R</span><span class="p">):</span>
<span class="c1"># Let's use the same random variables for all methods. This allows </span>
<span class="c1"># for a lower variance comparsion and equivalence testing.</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">logit</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="c1"># used in fast method: precompute expensive stuff.</span>
<span class="c1"># Requires computing sigmoid for each x.</span>
<span class="k">with</span> <span class="n">T</span><span class="p">[</span><span class="s1">'slow1'</span><span class="p">]:</span>
<span class="n">s1</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">></span> <span class="n">u</span>
<span class="c1"># Avoid memory allocation in slow-1 by using the out option to sigmoid</span>
<span class="c1"># function. It's a little bit faster than slow-1.</span>
<span class="k">with</span> <span class="n">T</span><span class="p">[</span><span class="s1">'slow2'</span><span class="p">]:</span>
<span class="n">sigmoid</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="n">tmp</span><span class="p">)</span>
<span class="n">s2</span> <span class="o">=</span> <span class="n">tmp</span> <span class="o">></span> <span class="n">u</span>
<span class="c1"># Rolling our sigmoid is a bit slower than using the library function.</span>
<span class="c1"># Not to mention this implementation isn't as numerically stable.</span>
<span class="k">with</span> <span class="n">T</span><span class="p">[</span><span class="s1">'slow3'</span><span class="p">]:</span>
<span class="n">s3</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">X</span><span class="p">))</span> <span class="o">></span> <span class="n">u</span>
<span class="c1"># The fast method.</span>
<span class="k">with</span> <span class="n">T</span><span class="p">[</span><span class="s1">'fast'</span><span class="p">]:</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">X</span> <span class="o">></span> <span class="n">z</span>
<span class="n">F</span> <span class="o">+=</span> <span class="n">f</span> <span class="o">/</span> <span class="n">R</span>
<span class="k">assert</span> <span class="p">(</span><span class="n">s1</span> <span class="o">==</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">all</span><span class="p">()</span>
<span class="k">assert</span> <span class="p">(</span><span class="n">s2</span> <span class="o">==</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">all</span><span class="p">()</span>
<span class="k">assert</span> <span class="p">(</span><span class="n">s3</span> <span class="o">==</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">all</span><span class="p">()</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">F</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">T</span><span class="o">.</span><span class="n">compare</span><span class="p">()</span>
</pre></div>
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<pre>fast is 26.8226x faster than slow1 <span class="ansi-yellow-fg">(avg: slow1: 0.000353234 fast: 1.31693e-05)</span>
slow2 is 1.0093x faster than slow1 <span class="ansi-yellow-fg">(avg: slow1: 0.000353234 slow2: 0.000349975)</span>
slow1 is 1.0920x faster than slow3 <span class="ansi-yellow-fg">(avg: slow3: 0.000385725 slow1: 0.000353234)</span>
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<p>It looks like our trick is about $28$x faster than the fastest competing slow method!</p>
<p>We also see that the assert statements passed, which means that the methods tested produce precisely the same samples.</p>
<p>The final plot demonstrates that we get the right expected value (red curve) as we sweep the distributions parameter (x-axis).</p>
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<h2 id="Bonus">Bonus<a class="anchor-link" href="#Bonus">¶</a></h2><p>We could alternatively use the <a href="http://timvieira.github.io/blog/post/2014/07/31/gumbel-max-trick/">Gumbel-max trick</a> to derive a similar algorithm. If we ground out the trick for a sigmoid instead of a general mutlinomal distributions, we end up with</p>
$$
Z_0 \sim \textrm{Gumbel}(0,1)
$$$$
Z_1 \sim \textrm{Gumbel}(0,1)
$$$$
Y = x > Z_0 - Z_1
$$<p>Much like our new trick, this one benefits from the fact that all expensive stuff is done independent of the data (i.e., the value of $x$). However, it seems silly that we "need" to generate <em>two</em> Gumbel RVs to get one sample from the sigmoid. With a little bit of Googling, we discover that the difference of $\textrm{Gumbel}(0,1)$ RVs is a <a href="https://en.wikipedia.org/wiki/Logistic_distribution">logistic</a> RV (specifically $\textrm{Logistic}(0,1)$).</p>
<p>It turns out that $\textrm{logit}(\textrm{Uniform}(0,1))$ is a $\textrm{Logistic}(0,1)$ RV.</p>
<p>Voila! Our fast sampling trick and the Gumbel-max trick are connected!</p>
<h2 id="Related-tricks">Related tricks<a class="anchor-link" href="#Related-tricks">¶</a></h2><p>Another trick is Justin Domke's <a href="https://justindomke.wordpress.com/2014/01/08/reducing-sigmoid-computations-by-at-least-88-0797077977882/">trick</a> to reduce calls to $\exp$ by $\approx 88\%$. The <em>disadvantage</em> of this approach is that it's harder to implement with vectorization. The <em>advantage</em> is that we don't need to precompute any expensive things.</p>
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<h2 id="Appendix">Appendix<a class="anchor-link" href="#Appendix">¶</a></h2>
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<h3 id="Logit-plot">Logit plot<a class="anchor-link" href="#Logit-plot">¶</a></h3>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
<span class="n">ys</span> <span class="o">=</span> <span class="n">logit</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">);</span>
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<h3 id="Logistic-random-variable">Logistic random variable<a class="anchor-link" href="#Logistic-random-variable">¶</a></h3><p>Check that our sampling method is equivalent to sampling from a logistic distribution.</p>
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<div class="prompt input_prompt">In [4]:</div>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">logistic</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">size</span><span class="o">=</span><span class="mi">10000</span><span class="p">)</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">logit</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">normed</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
<span class="n">ys</span> <span class="o">=</span> <span class="n">logistic</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">);</span>
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</script>Sqrt-biased sampling2016-06-28T00:00:00-04:002016-06-28T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2016-06-28:/blog/post/2016/06/28/sqrt-biased-sampling/<p>The following post is about instance of "sampling in proportion to <span class="math">\(p\)</span> is not
optimal, but you probably think it is." It's surprising how few people seem to
know this trick. Myself included! It was brought to my attention recently by
<a href="http://lowrank.net/nikos/">Nikos Karampatziakis</a>. (Thanks, Nikos!)</p>
<p>The paper credited for this trick is
<a href="http://www.pnas.org/content/106/6/1716.full.pdf">Press (2008)</a>. I'm borrowing
heavily from that paper as well as an email exchange from Nikos.</p>
<p><strong>Setting</strong>: Suppose you're an aspiring chef with a severe head injury affecting
your long- and short- term memory trying to find a special recipe from a
cookbook that you made one time but just can't remember exactly which recipe
it was. So, based on the ingredients of each recipe, you come up with a prior
probability <span class="math">\(p_i\)</span> that recipe <span class="math">\(i\)</span> is the one you're looking for. In total, the
cookbook has <span class="math">\(n\)</span> recipes and <span class="math">\(\sum_{i=1}^n p_i = 1.\)</span></p>
<p>A good strategy would be to sort recipes by <span class="math">\(p_i\)</span> and cook the most promising
ones first. Unfortunately, you're not a great chef so there is some probability
that you'll mess-up the recipe. So, it's a good idea to try recipes multiple
times. Also, you have no short term memory...</p>
<p>This suggests a <em>sampling with replacement</em> strategy, where we sample a recipe
from the cookbook to try <em>independently</em> of whether we've tried it before
(called a <em>memoryless</em> strategy). Let's give this strategy the name
<span class="math">\(\boldsymbol{q}.\)</span> Note that <span class="math">\(\boldsymbol{q}\)</span> is a probability distribution over
the recipes in the cookbook, just like <span class="math">\(\boldsymbol{p}.\)</span></p>
<p><strong>How many recipes until we find the special one?</strong> To start, suppose the
special recipe is <span class="math">\(j.\)</span> Then, the expected number of recipes we have to make
until we find <span class="math">\(j\)</span> under the strategy <span class="math">\(\boldsymbol{q}\)</span> is</p>
<div class="math">$$
\sum_{t=1}^\infty t \cdot (1 - q_j)^{t-1} q_{j} = 1/q_{j}.
$$</div>
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<p><button onclick="toggle('#derivation-series')" class="toggle-button">Derivation</button>
<div id="derivation-series" style="display:none;" class="derivation">
<strong>Derivation</strong>:</p>
<p>We start with
</p>
<div class="math">$$
\sum_{t=1}^\infty t \cdot (1 - q_j)^{t-1} q_{j},
$$</div>
<p>Let <span class="math">\(a = (1-q_j)\)</span>, to clean up notation.
</p>
<div class="math">$$
= q_{j} \sum_{t=1}^\infty t \cdot a^{t-1}
$$</div>
<p>Use the identity <span class="math">\(\nabla_a [ a^t ] = t \cdot a^{t-1}\)</span>,
</p>
<div class="math">$$
= q_{j} \sum_{t=1}^\infty \nabla_a[ a^{t} ].
$$</div>
<p>Fish the gradient out of the sum and tweak summation index,
</p>
<div class="math">$$
= q_{j} \nabla_a\left[ \sum_{t=1}^\infty a^{t} \right]
= q_{j} \nabla_a\left[ -1 + \sum_{t=0}^\infty a^{t}\right]
$$</div>
<p>Plugin in the solution to the geometric series,
</p>
<div class="math">$$
= q_{j} \nabla_a\left[ -1 + \frac{1}{1-a} \right].
$$</div>
<p>Take derivative, expand <span class="math">\(a\)</span> and simplify,
</p>
<div class="math">$$
= q_{j} \frac{1}{(1-a)^2}
= \frac{1}{q_j}
$$</div>
</div>
<p>The equation says that expected time it takes to sample <span class="math">\(j\)</span> for <em>the first time</em>
is the probability we didn't sample <span class="math">\(j\)</span> for <span class="math">\((t-1)\)</span> steps times the probability we
sample <span class="math">\(j\)</span> at time <span class="math">\(t.\)</span> We multiply this probability by the time <span class="math">\(t\)</span> to get the
<em>expected</em> time.</p>
<p>Note that this equation assumes that we know <span class="math">\(j\)</span> is the special recipe <em>with
certainty</em> when we sample it. We'll revisit this assumption later when we
consider potential errors in executing the recipe.</p>
<p>Since we don't known which <span class="math">\(j\)</span> is the right one, we take an expectation over it
according to the prior distribution, which yields the following equation,
</p>
<div class="math">$$
f(\boldsymbol{q}) = \sum_{i=1}^n \frac{p_i}{q_i}.
$$</div>
<p><strong>The first surprising thing</strong>: Uniform is just as good as <span class="math">\(\boldsymbol{p}\)</span>,
yikes! <span class="math">\(f(\boldsymbol{p}) = \sum_{i=1}^n \frac{p_i}{p_i} = n\)</span> and
<span class="math">\(f(\text{uniform}(n)) = \sum_{i=1}^n \frac{p_i }{ 1/n } = n.\)</span> (Assume, without
loss of generality, that <span class="math">\(p_i > 0\)</span> since we can just drop these elements from
<span class="math">\(\boldsymbol{p}.\)</span>)</p>
<p><strong>What's the <em>optimal</em> <span class="math">\(\boldsymbol{q}\)</span>?</strong> We can address this question by
solving the following optimization (which will have a nice closed form
solution),</p>
<div class="math">$$
\begin{eqnarray*}
&& \boldsymbol{q}^* = \underset{\boldsymbol{q}}{\operatorname{argmin}} \sum_{i=1}^n \frac{p_i}{q_i} \\
&& \ \ \ \ \ \ \ \ \text{ s.t. } \sum_{i=1}^n q_i = 1 \\
&& \ \ \ \ \ \ \ \ \ \ \ \ \, q_1 \ldots q_n \ge 0.
\end{eqnarray*}
$$</div>
<p>The optimization problem says minimize the expected time to find the special
recipe. The constraints enforce that <span class="math">\(\boldsymbol{q}\)</span> is a valid probability
distribution.</p>
<p>The optimal strategy, which we get via Lagrange multipliers, turns out to be,
</p>
<div class="math">$$
q^*_i = \frac{ \sqrt{p_i} }{ \sum_{j=1}^n \sqrt{p_j} }.
$$</div>
<p><button onclick="toggle('#Lagrange')" class="toggle-button">Derivation</button>
<div id="Lagrange" style="display:none;" class="derivation">
To solve this constrained optimization problem, we form the
Lagrangian,</p>
<div class="math">$$\mathcal{L}(\boldsymbol{q}, \lambda) = \sum_{i=1}^n \frac{p_i}{q_i} - \lambda\cdot \left(1 - \sum_{i=1}^n q_i\right),$$</div>
<p>and solve for <span class="math">\(\boldsymbol{q}\)</span> and multiplier <span class="math">\(\lambda\)</span> such that partial
derivatives are all equal to zero. This gives us the following system of
nonlinear equations,</p>
<div class="math">$$
\begin{eqnarray*}
&& \lambda - \frac{p_i}{q_i^2} = 0 \ \ \ \text{for } 1 \le i \le n \\
&& \lambda \cdot \left(1 - \sum_{i=1}^n q_i \right) = 0.
\end{eqnarray*}
$$</div>
<p>We see that <span class="math">\(q_i = \pm \sqrt{\frac{p_i}{\lambda}}\)</span> works for the first set of
equations, but since we need <span class="math">\(q_i \ge 0\)</span>, we take the positive one. Solving for
<span class="math">\(\lambda\)</span> and plugging it in, we get a normalized distribution,</p>
<div class="math">$$
q^*_i = \frac{ \sqrt{p_i} }{ \sum_{j=1}^n \sqrt{p_j} }.
$$</div>
</div>
<p><strong>How much better is <span class="math">\(q^*\)</span>?</strong>
</p>
<div class="math">$$
f(q^*) = \sum_i \frac{p_i}{q^*_i}
= \sum_i \frac{p_i}{ \frac{\sqrt{p_i} }{ \sum_j \sqrt{p_j}} }
= \left( \sum_i \frac{p_i}{ \sqrt{p_i} } \right) \left( \sum_j \sqrt{p_j} \right)
= \left( \sum_i \sqrt{p_i} \right)^2
$$</div>
<p>which sometimes equals <span class="math">\(n\)</span>, e.g., when <span class="math">\(\boldsymbol{p}\)</span> is uniform, but is never
bigger than <span class="math">\(n.\)</span></p>
<p><strong>What's the intuition?</strong> The reason why the <span class="math">\(\sqrt{p}\)</span>-scheme is preferred is
because we save on <em>additional</em> cooking experiments. For example, if a recipe
has <span class="math">\(k\)</span> times higher prior probability than the average recipe, then we will try
that recipe <span class="math">\(\sqrt{k}\)</span> times more often; compared to <span class="math">\(k\)</span>, which we'd get under
<span class="math">\(\boldsymbol{p}.\)</span> Additional cooking experiments are not so advantageous.</p>
<p><strong>Allowing for noise in the cooking process</strong>: Suppose that for each recipe we
had a prior belief about how hard that recipe is for us to cook. Denote that
belief <span class="math">\(s_i\)</span>, these beliefs are between zero (never get it right) and one
(perfect every time) and do not necessarily sum to one over the cookbook.</p>
<p>Following a similar derivation to before, the time to cook the special recipe
<span class="math">\(j\)</span> and cook it correctly is,
</p>
<div class="math">$$
\sum_{t=1}^\infty t \cdot (1 - \color{red}{s_j} q_j)^{t-1} q_{j} \color{red}{s_j} = \frac{1}{s_j \cdot q_j}
$$</div>
<p>
That gives rise to a modified objective,
</p>
<div class="math">$$
f'(\boldsymbol{q}) = \sum_{i=1}^n \frac{p_i}{\color{red}{s_i} \cdot q_i}
$$</div>
<p>This is exactly the same as the previous objective, except we've replaced <span class="math">\(p_i\)</span>
with <span class="math">\(p_i/s_i.\)</span> Thus, we can reuse our previous derivation to get the optimal
strategy, <span class="math">\(q^*_i \propto \sqrt{p_i / s_i}.\)</span> If noise is constant, then we
recover the original solution, <span class="math">\(q^*_i \propto \sqrt{p_i}.\)</span></p>
<p><strong>Extension to finding multiple tasty recipes</strong>: Suppose we're trying to find
several tasty recipes, not just a single special one. Now, <span class="math">\(p_i\)</span> is our prior
belief that we'll like the recipe at all. How do we minimize the time until we
find a tasty one? It turns out the same trick works without modification
because all derivations apply to each recipe independently. The same trick
works if <span class="math">\(p_i\)</span> does not sums to one over <span class="math">\(n.\)</span> For example, if <span class="math">\(p_i\)</span> is the
independent probability that you'll like recipe <span class="math">\(i\)</span> at all, not the
probability that it's the special one.</p>
<p><strong>Beyond memoryless policies</strong>: Clearly, our choice of a memoryless policy can
be beat by a policy family that balances exploration (trying new recipes) and
exploitation (trying our best guess).</p>
<ul>
<li>
<p>Overall, the problem we've posed is similar to a
<a href="https://en.wikipedia.org/wiki/Multi-armed_bandit">multi-armed bandit</a>. In
our case, the arms are the recipes, pulling the arm is trying the recipe and
the reward is whether or not we liked the recipe (possibly noisy). The key
difference between our setup and multi-armed bandits is that we trust our
prior distribution <span class="math">\(\boldsymbol{p}\)</span> and noise model <span class="math">\(\boldsymbol{s}.\)</span></p>
</li>
<li>
<p>If the amount of noise <span class="math">\(s_i\)</span> is known and we trust the prior <span class="math">\(p_i\)</span> then
there is an optimal deterministic (without-replacement) strategy that we can
get by sorting the recipes by <span class="math">\(p_i\)</span> accounting for the error rates
<span class="math">\(s_i.\)</span> This approach is described in the original paper.</p>
</li>
</ul>
<p><strong>A more realistic application</strong>: In certain language modeling applications, we
avoid computing normalization constants (which require summing over a massive
vocabulary) by using importance sampling, negative sampling or noise
contrastive estimation techniques (e.g.,
<a href="https://arxiv.org/pdf/1511.06909.pdf">Ji+,16</a>;
<a href="http://www.aclweb.org/anthology/Q15-1016">Levy+,15</a>). These techniques depend
on a proposal distribution, which folks often take to be the unigram
distribution. Unfortunately, this gives too many samples of stop words (e.g.,
"the", "an", "a"), so practitioners "anneal" the unigram distribution (to
increase the entropy), that is sample from <span class="math">\(q_i \propto
p_{\text{unigram},i}^\alpha.\)</span> Typically, <span class="math">\(\alpha\)</span> is set by grid search and
(no surprise) <span class="math">\(\alpha \approx 1/2\)</span> tends to work best! The <span class="math">\(\sqrt{p}\)</span>-sampling
trick is possibly a reverse-engineered justification in favor of annealing as
"the right thing to do" (e.g., why not do additive smoothing?) and it even
tells us how to set the annealing parameter <span class="math">\(\alpha.\)</span> The key assumption is
that we want to sample the actual word at a given position as often as
possible while still being diverse thanks to the coverage of unigram
prior. (Furthermore, memoryless sampling leads to simpler algorithms.)</p>
<!--
Actually, many word2vec papers use $\alpha=3/4$, which was suggested in
[Levy+,15](http://www.aclweb.org/anthology/Q15-1016), including the default
value in
[gensim](https://github.com/RaRe-Technologies/gensim/blob/develop/gensim/models/word2vec.py#L462). So,
[Ryan Cotterell](https://ryancotterell.github.io/) ran a quick experiment with
gensim, which confirmed the suspicion that $1/2$ may be better than $3/4.$
Word similarity accuracy (avg of 10 runs)
| alpha | accuracy |
+==================+
| 0.00 | 0.354 |
| 0.25 | 0.403 |
| 0.50 | 0.414 |
| 0.75 | 0.395 |
| 1.00 | 0.345 |
-->
<p><strong>Final remarks</strong>: I have uploaded a <a href="https://github.com/timvieira/blog/blob/master/content/notebook/Sqrt-biased-sampling.ipynb">Jupyter notebook</a> with test cases that illustrate the ideas in this article.</p>
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</script>The optimal proposal distribution is not p2016-05-28T00:00:00-04:002016-05-28T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2016-05-28:/blog/post/2016/05/28/the-optimal-proposal-distribution-is-not-p/<p>The following is a quick rant about
<a href="http://timvieira.github.io/blog/post/2014/12/21/importance-sampling/">importance sampling</a>
(see that post for notation).</p>
<p>I've heard the following <strong>incorrect</strong> statement one too many times,</p>
<blockquote>
<p>We chose <span class="math">\(q \approx p\)</span> because <span class="math">\(q=p\)</span> is the "optimal" proposal distribution.</p>
</blockquote>
<p>While it is certainly a good idea to pick <span class="math">\(q\)</span> to be as similar as possible to
<span class="math">\(p\)</span>, it is by no means <em>optimal</em> because it is oblivious to <span class="math">\(f\)</span>!</p>
<p>With importance sampling, it is possible to achieve a variance reduction over
Monte Carlo estimation. The optimal proposal distribution, assuming <span class="math">\(f(x) \ge 0\)</span>
for all <span class="math">\(x\)</span>, is <span class="math">\(q(x) \propto p(x) f(x).\)</span> This choice of <span class="math">\(q\)</span> gives us a <em>zero
variance</em> estimate <em>with a single sample</em>!</p>
<p>Of course, this is an unreasonable distribution to use because the normalizing
constant <em>is the thing you are trying to estimate</em>, but it is proof that <em>better
proposal distributions exist</em>.</p>
<p>The key to doing better than <span class="math">\(q=p\)</span> is to take <span class="math">\(f\)</span> into account. Look up
"importance sampling for variance reduction" to learn more.</p>
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</script>Dimensional analysis of gradient ascent2016-05-27T00:00:00-04:002016-05-27T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2016-05-27:/blog/post/2016/05/27/dimensional-analysis-of-gradient-ascent/<p>In physical sciences, numbers are paired with units and called quantities. In
this augmented number system, dimensional analysis provides a crucial sanity
check, much like type checking in a programming language. There are simple rules
for building up units and constraints on what operations are allowed. For
example, you can't multiply quantities which are not conformable or add
quantities with different units. Also, we generally know the units of the input
and desired output, which allows us to check that our computations at least
produce the right units.</p>
<p>In this post, we'll discuss the dimensional analysis of gradient ascent, which
will hopefully help us understand why the "step size" is parameter so finicky
and why it even exists.</p>
<p>Gradient ascent is an iterative procedure for (locally) maximizing a function,
<span class="math">\(f: \mathbb{R}^d \mapsto \mathbb{R}\)</span>.</p>
<div class="math">$$
x_{t+1} = x_t + \alpha \frac{\partial f(x_t)}{\partial x}
$$</div>
<p>In general, <span class="math">\(\alpha\)</span> is a <span class="math">\(d \times d\)</span> matrix, but often we constrain the matrix
to be simple, e.g., <span class="math">\(a\cdot I\)</span> for some scalar <span class="math">\(a\)</span> or <span class="math">\(\text{diag}(a)\)</span> for some
vector <span class="math">\(a\)</span>.</p>
<p>Now, let's look at the units of the change in <span class="math">\(\Delta x=x_{t+1} - x_t\)</span>,
</p>
<div class="math">$$
(\textbf{units }\Delta x) = \left(\textbf{units }\alpha\cdot \frac{\partial f(x_t)}{\partial x}\right) = (\textbf{units }\alpha) \frac{(\textbf{units }f)}{(\textbf{units }x)}.
$$</div>
<p>The units of <span class="math">\(\Delta x\)</span> must be <span class="math">\((\textbf{units }x)\)</span>. However, if we assume <span class="math">\(f\)</span>
is unit free, we're happy with <span class="math">\((\textbf{units }x) / (\textbf{units }f)\)</span>.</p>
<p>Solving for the units of <span class="math">\(\alpha\)</span> we get,
</p>
<div class="math">$$
(\textbf{units }\alpha) = \frac{(\textbf{units }x)^2}{(\textbf{units }f)}.
$$</div>
<p>This gives us an idea for what <span class="math">\(\alpha\)</span> should be.</p>
<p>For example, the inverse Hessian passes the unit check (if we assume <span class="math">\(f\)</span> unit
free). The disadvantages of the Hessian is that it needs to be positive-definite
(or at least invertible) in order to be a valid "step size" (i.e., we need
step sizes to be <span class="math">\(> 0\)</span>).</p>
<p>Another method for handling step sizes is line search. However, line search
won't let us run online. Furthermore, line search would be too slow in the case
where we want a step size for each dimension.</p>
<p>In machine learning, we've become fond of online methods, which adapt the step
size as they go. The general idea is to estimate a step size matrix that passes
the unit check (for each dimension of <span class="math">\(x\)</span>). Furthermore, we want do as little
extra work as possible to get this estimate (e.g., we want to avoid computing a
Hessian because that would be extra work). So, the step size should be based
only iterates and gradients up to time <span class="math">\(t\)</span>.</p>
<ul>
<li>
<p><a href="http://www.magicbroom.info/Papers/DuchiHaSi10.pdf">AdaGrad</a> doesn't doesn't
pass the unit check. This motivated AdaDelta.</p>
</li>
<li>
<p><a href="https://arxiv.org/abs/1212.5701">AdaDelta</a> uses the ratio of (running
estimates of) the root-mean-squares of <span class="math">\(\Delta x\)</span> and <span class="math">\(\partial f / \partial
x\)</span>. The mean is taken using an exponentially weighted moving average. See
paper for actual implementation.</p>
</li>
<li>
<p><a href="http://arxiv.org/abs/1412.6980">Adam</a> came later and made some tweaks to
remove (unintended) bias in the AdaDelta estimates of the numerator and
denominator.</p>
</li>
</ul>
<p>In summary, it's important/useful to analyze the units of numerical algorithms
in order to get a sanity check (i.e., catch mistakes) as well as to develop an
understanding of why certain parameters exist and how properties of a problem
affect the values we should use for them.</p>
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</script>Gradient-based hyperparameter optimization and the implicit function theorem2016-03-05T00:00:00-05:002016-03-05T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2016-03-05:/blog/post/2016/03/05/gradient-based-hyperparameter-optimization-and-the-implicit-function-theorem/<p>The most approaches to hyperparameter optimization can be viewed as a bi-level
optimization—the "inner" optimization optimizes training loss (wrt <span class="math">\(\theta\)</span>),
while the "outer" optimizes hyperparameters (<span class="math">\(\lambda\)</span>).</p>
<div class="math">$$
\lambda^* = \underset{\lambda}{\textbf{argmin}}\
\mathcal{L}_{\text{dev}}\left(
\underset{\theta}{\textbf{argmin}}\
\mathcal{L}_{\text{train}}(\theta, \lambda) \right)
$$</div>
<p>Can we estimate <span class="math">\(\frac{\partial \mathcal{L}_{\text{dev}}}{\partial \lambda}\)</span> so
that we can run gradient-based optimization over <span class="math">\(\lambda\)</span>?</p>
<p>Well, what does it mean to have an <span class="math">\(\textbf{argmin}\)</span> inside a function?</p>
<p>Well, it means that there is a <span class="math">\(\theta^*\)</span> that gets passed to
<span class="math">\(\mathcal{L}_{\text{dev}}\)</span>. And, <span class="math">\(\theta^*\)</span> is a function of <span class="math">\(\lambda\)</span>, denoted
<span class="math">\(\theta(\lambda)\)</span>. Furthermore, <span class="math">\(\textbf{argmin}\)</span> must set the derivative of the
inner optimization to zero in order to be a local optimum of the inner
function. So we can rephrase the problem as</p>
<div class="math">$$
\lambda^* = \underset{\lambda}{\textbf{argmin}}\
\mathcal{L}_{\text{dev}}\left(\theta(\lambda) \right),
$$</div>
<p>
where <span class="math">\(\theta(\lambda)\)</span> is the solution to
</p>
<div class="math">$$
\frac{\partial \mathcal{L}_{\text{train}}(\theta, \lambda)}{\partial \theta} = 0.
$$</div>
<p>Now how does <span class="math">\(\theta\)</span> change as the result of an infinitesimal change to
<span class="math">\(\lambda\)</span>?</p>
<p>The constraint on the derivative implies a type of "equilibrium"—the inner
optimization process will continue to optimize regardless of how we change
<span class="math">\(\lambda\)</span>. Assuming we don't change <span class="math">\(\lambda\)</span> too much, then the inner
optimization shouldn't change <span class="math">\(\theta\)</span> too much and it will change in a
predictable way.</p>
<p>To do this, we'll appeal to the implicit function theorem. Let's look at the
general case to simplify notation. Suppose <span class="math">\(x\)</span> and <span class="math">\(y\)</span> are related through a
function <span class="math">\(g\)</span> as follows,</p>
<div class="math">$$g(x,y) = 0.$$</div>
<p>Assuming <span class="math">\(g\)</span> is a smooth function in <span class="math">\(x\)</span> and <span class="math">\(y\)</span>, we can perturb either
argument, say <span class="math">\(x\)</span> by a small amount <span class="math">\(\Delta_x\)</span> and <span class="math">\(y\)</span> by <span class="math">\(\Delta_y\)</span>. Because
system preserves the constraint, i.e.,</p>
<div class="math">$$
g(x + \Delta_x, y + \Delta_y) = 0.
$$</div>
<p>We can solve for the change of <span class="math">\(x\)</span> as a result of an infinitesimal change in
<span class="math">\(y\)</span>. We take the first-order expansion,</p>
<div class="math">$$
g(x, y) + \Delta_x \frac{\partial g}{\partial x} + \Delta_y \frac{\partial g}{\partial y} = 0.
$$</div>
<p>Since <span class="math">\(g(x,y)\)</span> is already zero,</p>
<div class="math">$$
\Delta_x \frac{\partial g}{\partial x} + \Delta_y \frac{\partial g}{\partial y} = 0.
$$</div>
<p>Next, we solve for <span class="math">\(\frac{\Delta_x}{\Delta_y}\)</span>.</p>
<div class="math">$$
\begin{align}
\Delta_x \frac{\partial g}{\partial x} &= - \Delta_y \frac{\partial g}{\partial y} \\
\frac{\Delta_x}{\Delta_y} &= -\left( \frac{\partial g}{\partial x} \right)^{-1} \frac{\partial g}{\partial y}.
\end{align}
$$</div>
<p>Back to the original problem: Now we can use the implicit function theorem to
estimate how <span class="math">\(\theta\)</span> varies in <span class="math">\(\lambda\)</span> by plugging in <span class="math">\(g \mapsto
\frac{\partial \mathcal{L}_{\text{train}}}{\partial \theta}\)</span>, <span class="math">\(x \mapsto \theta\)</span>
and <span class="math">\(y \mapsto \lambda\)</span>:</p>
<div class="math">$$
\frac{\partial \theta}{\partial \lambda} = - \left( \frac{ \partial^2 \mathcal{L}_{\text{train}} }{ \partial \theta\, \partial \theta^\top } \right)^{-1} \frac{ \partial^2 \mathcal{L}_{\text{train}} }{ \partial \theta\, \partial \lambda^\top}
$$</div>
<p>This tells us how <span class="math">\(\theta\)</span> changes with respect to an infinitesimal change to
<span class="math">\(\lambda\)</span>. Now, we can apply the chain rule to get the gradient of the whole
optimization problem wrt <span class="math">\(\lambda\)</span>,</p>
<div class="math">$$
\frac{\partial \mathcal{L}_{\text{dev}}}{\partial \lambda}
= \frac{\partial \mathcal{L}_{\text{dev}}}{\partial \theta} \left( - \left( \frac{ \partial^2 \mathcal{L}_{\text{train}} }{ \partial \theta\, \partial \theta^\top } \right)^{-1} \frac{ \partial^2 \mathcal{L}_{\text{train}} }{ \partial \theta\, \partial \lambda^\top} \right)
$$</div>
<p>Since we don't like (explicit) matrix inverses, we compute <span class="math">\(- \left( \frac{
\partial^2 \mathcal{L}_{\text{train}} }{ \partial \theta\, \partial \theta^\top
} \right)^{-1} \frac{ \partial^2 \mathcal{L}_{\text{train}} }{ \partial \theta\,
\partial \lambda^\top}\)</span> as the solution to <span class="math">\(\left( \frac{ \partial^2
\mathcal{L}_{\text{train}} }{ \partial \theta\, \partial \theta^\top } \right) x
= -\frac{ \partial^2 \mathcal{L}_{\text{train}}}{ \partial \theta\, \partial
\lambda^\top}\)</span>. When the Hessian is positive definite, the linear system can be
solved with conjugate gradient, which conveniently only requires matrix-vector
products—i.e., you never have to materialize the Hessian. (Apparently,
<a href="https://en.wikipedia.org/wiki/Matrix-free_methods">matrix-free linear algebra</a>
is a thing.) In fact, you don't even have to implement the Hessian-vector and
Jacobian-vector products because they are accurately and efficiently
approximated with centered differences (see
<a href="/blog/post/2014/02/10/gradient-vector-product/">earlier post</a>).</p>
<p>At the end of the day, this is an easy algorithm to implement! However, the
estimate of the gradient can be temperamental if the linear system is
ill-conditioned.</p>
<p>In a later post, I'll describe a more-robust algorithms based on automatic
differentiation through the inner optimization algorithm, which make fewer and
less-brittle assumptions about the inner optimization.</p>
<p><strong>Further reading</strong>:</p>
<ul>
<li>
<p><a href="https://justindomke.wordpress.com/2014/02/03/truncated-bi-level-optimization/">Truncated Bi-Level Optimization</a></p>
</li>
<li>
<p><a href="http://ai.stanford.edu/~chuongdo/papers/learn_reg.pdf">Efficient multiple hyperparameter learning for log-linear models</a></p>
</li>
<li>
<p><a href="http://arxiv.org/abs/1502.03492">Gradient-based Hyperparameter Optimization through Reversible Learning</a></p>
</li>
<li>
<p><a href="http://fa.bianp.net/blog/2016/hyperparameter-optimization-with-approximate-gradient/">Hyperparameter optimization with approximate gradient</a>
(<a href="https://arxiv.org/pdf/1602.02355.pdf">paper</a>): This paper looks at the implicit
differentiation approach where you have an <em>approximate</em>
solution to the inner optimization problem. They are able to provide error bounds and
convergence guarantees under some reasonable conditions.</p>
</li>
</ul>
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</script>Multidimensional array index2016-01-17T00:00:00-05:002016-01-17T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2016-01-17:/blog/post/2016/01/17/multidimensional-array-index/<p>This is a simple note on how to compute a bijective mapping between the indices
of an <span class="math">\(n\)</span>-dimensional array and a flat, one-dimensional array. We'll look at
both directions of the mapping: <code>(tuple->int)</code> and <code>(int -> tuple)</code>.</p>
<p>We'll assume each dimension <span class="math">\(a, b, c, \ldots\)</span> is a positive integer and bounded
<span class="math">\(a \le A, b \le B, c \le C, \ldots\)</span></p>
<h3>Start small</h3>
<p>Let's start by looking at <span class="math">\(n = 3\)</span> and generalize from there.</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">index_3</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
<span class="n">_</span><span class="p">,</span><span class="n">J</span><span class="p">,</span><span class="n">K</span> <span class="o">=</span> <span class="n">A</span>
<span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">,</span><span class="n">k</span> <span class="o">=</span> <span class="n">a</span>
<span class="k">return</span> <span class="p">((</span><span class="n">i</span><span class="o">*</span><span class="n">J</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span><span class="o">*</span><span class="n">K</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">inverse_3</span><span class="p">(</span><span class="n">ix</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
<span class="n">_</span><span class="p">,</span><span class="n">J</span><span class="p">,</span><span class="n">K</span> <span class="o">=</span> <span class="n">A</span>
<span class="n">total</span> <span class="o">=</span> <span class="n">J</span><span class="o">*</span><span class="n">K</span>
<span class="n">i</span> <span class="o">=</span> <span class="n">ix</span> <span class="o">//</span> <span class="n">total</span>
<span class="n">ix</span> <span class="o">=</span> <span class="n">ix</span> <span class="o">%</span> <span class="n">total</span>
<span class="n">total</span> <span class="o">=</span> <span class="n">K</span>
<span class="n">j</span> <span class="o">=</span> <span class="n">ix</span> <span class="o">//</span> <span class="n">total</span>
<span class="n">k</span> <span class="o">=</span> <span class="n">ix</span> <span class="o">%</span> <span class="n">total</span>
<span class="k">return</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">,</span><span class="n">k</span><span class="p">)</span>
</code></pre></div>
<p>Here's our test case:</p>
<div class="highlight"><pre><span></span><code><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">C</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span>
<span class="n">key</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">A</span><span class="p">):</span>
<span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B</span><span class="p">):</span>
<span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">C</span><span class="p">):</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">),</span> <span class="s1">'->'</span><span class="p">,</span> <span class="n">key</span>
<span class="k">assert</span> <span class="n">inverse_3</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">C</span><span class="p">))</span> <span class="o">==</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">index_3</span><span class="p">((</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">),</span> <span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">C</span><span class="p">))</span> <span class="o">==</span> <span class="n">key</span>
<span class="n">key</span> <span class="o">+=</span> <span class="mi">1</span>
</code></pre></div>
<p>Note: This is not the only bijective mapping from <code>tuple</code> to <code>int</code> that we
could have come up with. The one we chose corresponds to the particular layout,
which is apparent in the test case.</p>
<p>For <span class="math">\(n=4\)</span> the pattern is <span class="math">\(((a \cdot B + b) \cdot C + d) \cdot D + d\)</span>.</p>
<p>Sidenote: We don't actually need the bound <span class="math">\(a \le A\)</span> in either <code>index</code> or
<code>inverse</code>. This gives us a little extra flexibility because our first
dimension can be infinite/unknown.</p>
<h3>General case</h3>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">index</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
<span class="s2">"Map tuple ``a`` to index with known bounds ``A``."</span>
<span class="c1"># the pattern:</span>
<span class="c1"># ((i*J + j)*K + k)*L + l</span>
<span class="n">key</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">)):</span>
<span class="n">key</span> <span class="o">*=</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="n">key</span> <span class="o">+=</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="k">return</span> <span class="n">key</span>
<span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="n">ix</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
<span class="s2">"Find key given index ``ix`` and bounds ``A``."</span>
<span class="n">total</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">A</span><span class="p">:</span>
<span class="n">total</span> <span class="o">*=</span> <span class="n">x</span>
<span class="n">key</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">)):</span>
<span class="n">total</span> <span class="o">/=</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="n">r</span> <span class="o">=</span> <span class="n">ix</span> <span class="o">//</span> <span class="n">total</span>
<span class="n">ix</span> <span class="o">=</span> <span class="n">ix</span> <span class="o">%</span> <span class="n">total</span>
<span class="n">key</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
<span class="k">return</span> <span class="n">key</span>
</code></pre></div>
<h2>Appendix</h2>
<h3>Testing the general case</h3>
<div class="highlight"><pre><span></span><code><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span><span class="o">,</span> <span class="nn">itertools</span>
<span class="k">def</span> <span class="nf">test_layout</span><span class="p">(</span><span class="n">D</span><span class="p">):</span>
<span class="s2">"Test that `index` produces the layout we expect."</span>
<span class="n">z</span> <span class="o">=</span> <span class="p">[</span><span class="n">index</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">D</span><span class="p">))]</span>
<span class="k">assert</span> <span class="n">z</span> <span class="o">==</span> <span class="nb">range</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="n">D</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">test_inverse</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">D</span><span class="p">):</span>
<span class="n">got</span> <span class="o">=</span> <span class="n">inverse</span><span class="p">(</span><span class="n">index</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">D</span><span class="p">),</span> <span class="n">D</span><span class="p">)</span>
<span class="k">assert</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="o">==</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">got</span><span class="p">)</span>
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
<span class="n">test_layout</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
<span class="n">test_layout</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">])</span>
<span class="n">test_layout</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="n">test_layout</span><span class="p">([</span><span class="mi">3</span><span class="p">])</span>
<span class="n">test_inverse</span><span class="p">(</span><span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,),</span> <span class="n">D</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,))</span>
<span class="n">test_inverse</span><span class="p">(</span><span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">D</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
<span class="n">test_inverse</span><span class="p">(</span><span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span> <span class="n">D</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
<span class="n">test_inverse</span><span class="p">(</span><span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">D</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
<span class="n">test_inverse</span><span class="p">(</span><span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">11</span><span class="p">),</span> <span class="n">D</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">20</span><span class="p">))</span>
</code></pre></div>
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</script>Gradient of a product2015-07-29T00:00:00-04:002015-07-29T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2015-07-29:/blog/post/2015/07/29/gradient-of-a-product/<div class="math">$$
\newcommand{\gradx}[1]{\grad{x}{ #1 }}
\newcommand{\grad}[2]{\nabla_{\! #1}\! \left[ #2 \right]}
\newcommand{\R}{\mathbb{R}}
\newcommand{\bigo}[0]{\mathcal{O}}
$$</div>
<p>In this post we'll look at how to compute the gradient of a product. This is
such a common subroutine in machine learning that it's worth careful
consideration. In a later post, I'll describe the gradient of a
sum-over-products, which is another interesting and common pattern in machine
learning (e.g., exponential families, CRFs, context-free grammar, case-factor
diagrams, semiring-weighted logic programming).</p>
<p>Given a collection of functions with a common argument <span class="math">\(f_1, \cdots, f_n \in \{
\R^d \mapsto \R \}\)</span>.</p>
<p>Define their product <span class="math">\(p(x) = \prod_{i=1}^n f_i(x)\)</span></p>
<p>Suppose, we'd like to compute the gradient of the product of these functions
with respect to their common argument, <span class="math">\(x\)</span>.</p>
<div class="math">$$
\begin{eqnarray*}
\gradx{ p(x) }
&=& \gradx{ \prod_{i=1}^n f_i(x) }
&=& \sum_{i=1}^n \left( \gradx{f_i(x)} \prod_{i \ne j} f_j(x) \right)
\end{eqnarray*}
$$</div>
<p>As you can see in the equation above, the gradient takes the form of a
"leave-one-out product" sometimes called a "cavity."</p>
<p>A naive method for computing the gradient computes the leave-one-out products
from scratch for each <span class="math">\(i\)</span> (outer loop)—resulting in a overall runtime of
<span class="math">\(O(n^2)\)</span> to compute the gradient. Later, we'll see a dynamic program for
computing this efficiently.</p>
<p><strong>Division trick</strong>: Before going down the dynamic programming rabbit hole, let's
consider the following relatively simple method for computing the gradient,
which uses division:</p>
<div class="math">$$
\begin{eqnarray*}
\gradx{ p(x) }
&=& \sum_{i=1}^n \left( \frac{\gradx{f_i(x)} }{ f_i(x) } \prod_{j=1}^n f_j(x) \right)
&=& \left( \sum_{i=1}^n \frac{\gradx{f_i(x)} }{ f_i(x) } \right) \left( \prod_{j=1}^n f_j(x) \right)
\end{eqnarray*}
$$</div>
<p>Pro:</p>
<ul>
<li>Runtime <span class="math">\(\bigo(n)\)</span> with space <span class="math">\(\bigo(1)\)</span>.</li>
</ul>
<p>Con:</p>
<ul>
<li>
<p>Requires <span class="math">\(f \ne 0\)</span>. No worries, we can handle zeros with three cases: (1) If
no zeros: the division trick works fine. (2) Only one zero: implies that only
one term in the sum will have a nonzero gradient, which we compute via
leave-one-out product. (3) Two or more zeros: all gradients are zero and
there is no work to be done.</p>
</li>
<li>
<p>Requires multiplicative inverse operator (division) <em>and</em>
associative-commutative multiplication, which means it's not applicable to
matrices.</p>
</li>
</ul>
<p><strong>Log trick</strong>: Suppose <span class="math">\(f_i\)</span> are very small numbers (e.g., probabilities), which
we'd rather not multiply together because we'll quickly lose precision (e.g.,
for large <span class="math">\(n\)</span>). It's common practice (especially in machine learning) to replace
<span class="math">\(f_i\)</span> with <span class="math">\(\log f_i\)</span>, which turns products into sums, <span class="math">\(\prod_{j=1}^n f_j(x) =
\exp \left( \sum_{j=1}^n \log f_j(x) \right)\)</span>, and tiny numbers (like
<span class="math">\(\texttt{3.72e-44}\)</span>) into large ones (like <span class="math">\(\texttt{-100}\)</span>).</p>
<p>Furthermore, using the identity <span class="math">\((\nabla g) = g \cdot \nabla \log g\)</span>, we can
operate exclusively in the "<span class="math">\(\log\)</span>-domain".</p>
<div class="math">$$
\begin{eqnarray*}
\gradx{ p(x) }
&=& \left( \sum_{i=1}^n \gradx{ \log f_i(x) } \right) \exp\left( \sum_{j=1}^n \log f_j(x) \right)
\end{eqnarray*}
$$</div>
<p>Pro:</p>
<ul>
<li>
<p>Numerically stable</p>
</li>
<li>
<p>Runtime <span class="math">\(\bigo(n)\)</span> with space <span class="math">\(\bigo(1)\)</span>.</p>
</li>
<li>
<p>Doesn't require multiplicative inverse assuming you can compute <span class="math">\(\gradx{ \log
f_i(x) }\)</span> without it.</p>
</li>
</ul>
<p>Con:</p>
<ul>
<li>
<p>Requires <span class="math">\(f > 0\)</span>. But, we can use
<a href="http://timvieira.github.io/blog/post/2015/02/01/log-real-number-class/">LogReal number class</a>
to represent negative numbers in log-space, but we still need to be careful
about zeros (like in the division trick).</p>
</li>
<li>
<p>Doesn't easily generalize to other notions of multiplication.</p>
</li>
</ul>
<p><strong>Dynamic programming trick</strong>: <span class="math">\(\bigo(n)\)</span> runtime and <span class="math">\(\bigo(n)\)</span> space. You may
recognize this as forward-backward algorithm for linear chain CRFs
(cf. <a href="http://www.inference.phy.cam.ac.uk/hmw26/papers/crf_intro.pdf">Wallach (2004)</a>,
section 7).</p>
<p>The trick is very straightforward when you think about it in isolation. Compute
the products of all prefixes and suffixes. Then, multiply them together.</p>
<p>Here are the equations:</p>
<div class="math">$$
\begin{eqnarray*}
\alpha_0(x) &=& 1 \\
\alpha_t(x)
&=& \prod_{i \le t} f_i(x)
= \alpha_{t-1}(x) \cdot f_t(x) \\
\beta_{n+1}(x) &=& 1 \\
\beta_t(x)
&=& \prod_{i \ge t} f_i(x) = f_t(x) \cdot \beta_{t+1}(x)\\
\gradx{ p(x) }
&=& \sum_{i=1}^n \left( \prod_{j < i} f_j(x) \right) \gradx{f_i(x)} \left( \prod_{j > i} f_j(x) \right) \\
&=& \sum_{i=1}^n \alpha_{i-1}(x) \cdot \gradx{f_i(x)} \cdot \beta_{i+1}(x)
\end{eqnarray*}
$$</div>
<p>Clearly, this requires <span class="math">\(O(n)\)</span> additional space.</p>
<p>Only requires an associative operator (i.e., Does not require it to be
commutative or invertible like earlier strategies).</p>
<p>Why do we care about the non-commutative multiplication? A common example is
matrix multiplication where <span class="math">\(A B C \ne B C A\)</span>, even if all matrices have the
conformable dimensions.</p>
<p><strong>Connections to automatic differentiation</strong>: The theory behind reverse-mode
automatic differentiation says that if you can compute a function, then you
<em>can</em> compute it's gradient with the same asymptotic complexity, <em>but</em> you might
need more space. That's exactly what we did here: We started with a naive
algorithm for computing the gradient with <span class="math">\(\bigo(n^2)\)</span> time and <span class="math">\(\bigo(1)\)</span> space
(other than the space to store the <span class="math">\(n\)</span> functions) and ended up with a <span class="math">\(\bigo(n)\)</span>
time <span class="math">\(\bigo(n)\)</span> space algorithm with a little clever thinking. What I'm saying
is autodiff—even if you don't use a magical package—tells us that an
efficient algorithm for the gradient always exists. Furthermore, it tells you
how to derive it manually, if you are so inclined. The key is to reuse
intermediate quantities (hence the increase in space).</p>
<p><em>Sketch</em>: In the gradient-of-a-product case, assuming we implemented
multiplication left-to-right (forward pass) that already defines the prefix
products (<span class="math">\(\alpha\)</span>). It turns out that the backward pass gives us <span class="math">\(\beta\)</span> as
adjoints. Lastly, we'd propagate gradients through the <span class="math">\(f\)</span>'s to get
<span class="math">\(\frac{\partial p}{\partial x}\)</span>. Essentially, we end up with exactly the dynamic
programming algorithm we came up with.</p>
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</script>Multiclass logistic regression and conditional random fields are the same thing2015-04-29T00:00:00-04:002015-04-29T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2015-04-29:/blog/post/2015/04/29/multiclass-logistic-regression-and-conditional-random-fields-are-the-same-thing/<p>A short rant: Multiclass logistic regression and conditional random fields (CRF)
are the same thing. This comes to a surprise to many people because CRFs tend to
be surrounded by additional "stuff."</p>
<p>Understanding this very basic connection not only deepens our understanding, but
also suggests a method for testing complex CRF code.</p>
<p>Multiclass logistic regression is simple. The goal is to predict the correct
label <span class="math">\(y^*\)</span> from handful of labels <span class="math">\(\mathcal{Y}\)</span> given the observation <span class="math">\(x\)</span> based
on features <span class="math">\(\phi(x,y)\)</span>. Training this model typically requires computing the
gradient:</p>
<div class="math">$$
\nabla \log p(y^* \mid x) = \phi(x,y^*) - \sum_{y \in \mathcal{Y}} p(y|x) \phi(x,y)
$$</div>
<p>where
</p>
<div class="math">$$
\begin{eqnarray*}
p(y|x) &=& \frac{1}{Z(x)} \exp(\theta^\top \phi(x,y)) & \ \ \ \ \text{and} \ \ \ \ &
Z(x) &=& \sum_{y \in \mathcal{Y}} \exp(\theta^\top \phi(x,y))
\end{eqnarray*}
$$</div>
<p>At test-time, we often take the highest-scoring label under the model.</p>
<div class="math">$$
\widehat{y}(x) = \underset{y \in \mathcal{Y}}{\textrm{argmax}}\ \theta^\top \phi(x,y)
$$</div>
<p>A conditional random field is <em>exactly</em> multiclass logistic regression. The only
difference is that the sums (<span class="math">\(Z(x)\)</span> and <span class="math">\(\sum_{y \in \mathcal{Y}} p(y|x)
\phi(x,y)\)</span>) and the argmax <span class="math">\(\widehat{y}(x)\)</span> are inefficient to compute naively
(i.e., by brute-force enumeration). This point is often lost when people first
learn about CRFs. Some people never make this connection.</p>
<p>Brute-force enumeration is a very useful method for testing complex dynamic
programming procedures for computing the sums and the argmax on relatively small
examples. Don't just copy code for dynamic programming out of a paper! Test it!</p>
<p>Here's some stuff you'll see once we start talking about CRFs:</p>
<ol>
<li>
<p>Inference algorithms (e.g., Viterbi decoding, forward-backward, Junction
tree)</p>
</li>
<li>
<p>Graphical models (factor graphs, Bayes nets, Markov random fields)</p>
</li>
<li>
<p>Model templates (i.e., repeated feature functions)</p>
</li>
</ol>
<p>In the logistic regression case, we'd never use the term "inference" to describe
the "sum" and "max" over a handful of categories. Once we move to a structured
label space, this term gets throw around. (BTW, this isn't "statistical
inference," just algorithms to compute sum and max over <span class="math">\(\mathcal{Y}\)</span>.)</p>
<p>Graphical models establish a notation and structural properties which allow
efficient inference—things like cycles and treewidth.</p>
<p>Model templating is the only essential trick to move from logistic regression to
a CRF. Templating "solves" the problem that not all training examples have the
same "size"—the set of outputs <span class="math">\(\mathcal{Y}(x)\)</span> now depends on <span class="math">\(x\)</span>. A model
template specifies how to compute the features for an entire output, by looking
at interactions between subsets of variables.</p>
<div class="math">$$
\phi(x,\boldsymbol{y}) = \sum_{\alpha \in A(x)} \phi_\alpha(x,
\boldsymbol{y}_\alpha)
$$</div>
<p>where <span class="math">\(\alpha\)</span> is a labeled subset of variables often called a factor and
<span class="math">\(\boldsymbol{y}_\alpha\)</span> is the subvector containing values of variables
<span class="math">\(\alpha\)</span>. Basically, the feature function <span class="math">\(\phi\)</span> gets to look at some subset of
the variables being predicted <span class="math">\(y\)</span> and the entire input <span class="math">\(x\)</span>. The ability to look
at more of <span class="math">\(y\)</span> allows the model to make more coherent predictions.</p>
<p>Anywho, it's often useful to take a step back and think about what you are
trying to compute instead of how you're computing it. In this post, this allowed
us see the similarity between logistic regression and CRFs even though they seem
quite different.</p>
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deep learning might not be as crazy as it seems. And maybe even convince some
deep learning lovers that the graphical models might have interesting things to
offer.</p>
<p>In the world of structured prediction, we are plagued by the high-treewidth
problem -- models with loopy factors are "bad" because exact inference is
intractable. There are three common approaches for dealing with this problem:</p>
<ol>
<li>
<p>Limit the expressiveness of the model (i.e., don't use to model you want)</p>
</li>
<li>
<p>Change the training objective</p>
</li>
<li>
<p>Approximate inference</p>
</li>
</ol>
<p>Approximate inference is tricky. Things can easily go awry.</p>
<p>For example, structured perceptron training with loopy max-product BP instead of
exact max product can diverge
<a href="http://papers.nips.cc/paper/3162-structured-learning-with-approximate-inference.pdf">(Kulesza & Pereira, 2007)</a>. Another
example: using approximate marginals from sum-product loopy BP in place of the
true marginals in the gradient of the log-likelihood. This results in a
different nonconvex objective function. (Note:
<a href="http://aclweb.org/anthology/C/C12/C12-1122.pdf">sometimes</a> these loopy BP
approximations work fine.)</p>
<p>It looks like using approximate inference during training changes the training
objective.</p>
<p>So, here's a simple idea: learn a model which makes accurate predictions given
the approximate inference algorithm that will be used at test-time. Furthermore,
we should minimize empirical risk instead of log-likelihood because it is robust
to model miss-specification and approximate inference. In other words, make
training conditions as close as possible to test-time conditions.</p>
<p>Now, as long as everything is differentiable, you can apply automatic
differentiation (backprop) to train the end-to-end system. This idea appears in
a few publications, including a handful of papers by Justin Domke, and a few by
Stoyanov & Eisner.</p>
<p>Unsuprisingly, it works pretty well.</p>
<p>I first saw this idea in
<a href="http://proceedings.mlr.press/v15/stoyanov11a/stoyanov11a.pdf">Stoyanov & Eisner (2011)</a>. They
use loopy belief propagation as their approximate inference algorithm. At the
end of the day, their model is essentially a deep recurrent network, which came
from unrolling inference in a graphical model. This idea really struck me
because it's clearly right in the middle between graphical models and deep
learning.</p>
<p>You can immediately imagine swapping in other approximate inference algorithms
in place of loopy BP.</p>
<p>Deep learning approaches get a bad reputation because there are a lot of
"tricks" to get nonconvex optimization to work and because model structures are
more open ended. Unlike graphical models, deep learning models have more
variation in model structures. Maybe being more open minded about model
structures is a good thing. We seem to have hit a brick wall with
likelihood-based training. At the same time, maybe we can port over some of the
good work on approximate inference as deep architectures.</p>Log-Real number class2015-02-01T00:00:00-05:002015-02-01T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2015-02-01:/blog/post/2015/02/01/log-real-number-class/<p>Most people know how to avoid numerical underflow in probability computations by
representing intermediate quantities in the log-domain. This trick turns
"multiplication" into "addition", "addition" into "logsumexp", "0" into
<span class="math">\(-\infty\)</span> and "1" into <span class="math">\(0\)</span>. Most importantly, it turns really small numbers into
reasonable-size numbers.</p>
<p>Unfortunately, without modification, this trick is limited to positive numbers
because <code>log</code> of a negative number is <code>NaN</code>.</p>
<p>Well, there is good news! For the cost of an extra bit, we can extend this trick
to the negative reals and furthermore, we get a bonafide ring instead of a mere
semiring.</p>
<p>I first saw this trick in
<a href="http://www.aclweb.org/anthology/D09-1005">Li and Eisner (2009)</a>. The trick is
nicely summarized in Table 3 of that paper, which I've pasted below.</p>
<div style="text-align:center">
<img src="/blog/images/logreal.png"/>
</div>
<p><strong>Why do I care?</strong> When computing gradients (e.g., gradient of risk),
intermediate values are rarely all positive. Furthermore, we're often
multiplying small things together. I've recently found log-reals to be effective
at squeaking a bit more numerical accuracy.</p>
<p>This trick is useful for almost all backprop computations because backprop is
essentially:</p>
<div class="highlight"><pre><span></span><code>adjoint(u) += adjoint(v) * dv/du.
</code></pre></div>
<p>The only tricky bit is lifting all <code>du/dv</code> computations into the log-reals.</p>
<p>Implementation:</p>
<ul>
<li>
<p>This trick is better suited to programming languages with structs. Using
objects will probably in an horrible slow down and using parallel arrays to
store the sign bit and double is probably too tedious and error prone. (Sorry
java folks.)</p>
</li>
<li>
<p>Here's a <a href="https://github.com/andre-martins/TurboParser/blob/master/src/util/logval.h">C++ implementation</a>
with operator overloading from Andre Martins</p>
</li>
<li>
<p>Note that log-real <code>+=</code> involves calls to <code>log</code> and <code>exp</code>, which will
definitely slow your code down a bit (these functions are much slower than
addition).</p>
</li>
</ul>
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</script>Importance sampling2014-12-21T00:00:00-05:002014-12-21T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2014-12-21:/blog/post/2014/12/21/importance-sampling/<p>Importance sampling is a powerful and pervasive technique in statistics, machine learning, and randomized algorithms.</p>
<h2>Basics</h2>
<p>Importance sampling is a technique for estimating the expectation <span class="math">\(\mu\)</span> of a random variable <span class="math">\(f(x)\)</span> under distribution <span class="math">\(p\)</span> from samples of a different distribution <span class="math">\(q.\)</span></p>
<p>The key observation is that <span class="math">\(\mu\)</span> can be expressed as the expectation of a different random variable <span class="math">\(f^*(x)=\frac{p(x)}{q(x)}\! \cdot\! f(x)\)</span> under <span class="math">\(q.\)</span></p>
<div class="math">$$
\mathbb{E}_{q}\! \left[ f^*(x) \right] = \mathbb{E}_{q}\! \left[ \frac{p(x)}{q(x)} f(x) \right] = \sum_{x} q(x) \frac{p(x)}{q(x)} f(x) = \sum_{x} p(x) f(x) = \mathbb{E}_{p}\! \left[ f(x) \right] = \mu
$$</div>
<p><br/>Technical condition: <span class="math">\(q\)</span> must have support everywhere <span class="math">\(p\)</span> does, <span class="math">\(f(x) p(x) > 0 \Rightarrow q(x) > 0.\)</span> Without this condition, the equation is biased! Note: <span class="math">\(q\)</span> can support things that <span class="math">\(p\)</span> doesn't.</p>
<p>Terminology: The quantity <span class="math">\(w(x) = \frac{p(x)}{q(x)}\)</span> is often referred to as the <em>importance weight</em> or <em>importance correction</em>. We often refer to <span class="math">\(p\)</span> as the target density and <span class="math">\(q\)</span> as the proposal density.</p>
<p>Now, given samples <span class="math">\(\{ x^{(i)} \}_{i=1}^{n}\)</span> from <span class="math">\(q,\)</span> we can use the Monte Carlo estimate, <span class="math">\(\hat{\mu} \approx \frac{1}{n} \sum_{i=1}^n f^{*}(x^{(i)}),\)</span> as an unbiased estimator of <span class="math">\(\mu.\)</span></p>
<h2>Remarks</h2>
<p>There are a few reasons we might want to use importance sampling:</p>
<ol>
<li>
<p><strong>Convenience</strong>: It might be trickier to sample directly from <span class="math">\(p.\)</span></p>
</li>
<li>
<p><strong>Bias-correction</strong>: Suppose we're developing an algorithm that requires samples to satisfy some safety condition (e.g., a minimum support threshold) and be unbiased. Importance sampling can be used to remove bias while satisfying the condition.</p>
</li>
<li>
<p><strong>Variance reduction</strong>: It might be the case that sampling directly from <span class="math">\(p\)</span> would require more samples to estimate <span class="math">\(\mu.\)</span> Check out these <a href="http://www.columbia.edu/~mh2078/MCS04/MCS_var_red2.pdf">great notes</a> for more.</p>
</li>
<li>
<p><strong>Off-policy evaluation and learning</strong>: We might want to collect some exploratory data from <span class="math">\(q\)</span> and evaluate different <em>policies</em> <span class="math">\(p\)</span> (e.g., to pick the best one). Here's a link to a future post on [off-policy evaluation and counterfactual reasoning (https://timvieira.github.io/blog/post/2016/12/19/counterfactual-reasoning-and-learning-from-logged-data/) and some cool papers:
<a href="http://arxiv.org/abs/1209.2355">counterfactual reasoning</a>,
<a href="http://arxiv.org/abs/cs/0204043">reinforcement learning</a>,
<a href="http://arxiv.org/abs/1103.4601">contextual bandits</a>,
<a href="http://papers.nips.cc/paper/4156-learning-bounds-for-importance-weighting.pdf">domain adaptation</a>.</p>
</li>
</ol>
<p>There are a few common cases for <span class="math">\(q\)</span> worth separate consideration:</p>
<ol>
<li>
<p><strong>Control over <span class="math">\(q\)</span></strong>: This is the case in experimental design, variance reduction, active learning, and reinforcement learning. It's often difficult to design <span class="math">\(q,\)</span> which results in an estimator with reasonable variance. A very difficult case is in off-policy evaluation because it (essentially) requires a good exploratory distribution for every possible policy. (I have much more to say on this topic.)</p>
</li>
<li>
<p><strong>Little to no control over <span class="math">\(q\)</span></strong>: For example, you're given some dataset (e.g., new articles), and you want to estimate performance on a different dataset (e.g., Twitter).</p>
</li>
<li>
<p><strong>Unknown <span class="math">\(q\)</span></strong>: In this case, we want to estimate <span class="math">\(q\)</span> (typically referred to as the propensity score) and use it in the importance sampling estimator. As far as I can tell, this technique is widely used to remove selection bias when estimating the effects of different treatments.</p>
</li>
</ol>
<p><strong>Drawbacks</strong>: The main drawback of importance sampling is variance. A few bad samples with large weights can drastically throw off the estimator. Thus, it's often the case that a biased estimator is preferred, e.g., <a href="https://hips.seas.harvard.edu/blog/2013/01/14/unbiased-estimators-of-partition-functions-are-basically-lower-bounds/">estimating the partition function</a>, <a href="http://arxiv.org/abs/1209.2355">clipping weights</a>, <a href="http://arxiv.org/abs/cs/0204043">indirect importance sampling</a>. A secondary drawback is that both densities must be normalized, which is often intractable.</p>
<p><strong>What's next?</strong> I plan to cover variance reduction and
<a href="https://timvieira.github.io/blog/post/2016/12/19/counterfactual-reasoning-and-learning-from-logged-data/">off-policy evaluation</a> in more detail in future posts.</p>
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</script>Numerically stable p-norms2014-11-10T00:00:00-05:002014-11-10T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2014-11-10:/blog/post/2014/11/10/numerically-stable-p-norms/<p>Consider the p-norm
</p>
<div class="math">$$
|| \boldsymbol{x} ||_p = \left( \sum_i |x_i|^p \right)^{\frac{1}{p}}
$$</div>
<p>In python this translates to:</p>
<div class="highlight"><pre><span></span><code><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">array</span>
<span class="k">def</span> <span class="nf">norm1</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
<span class="s2">"First-pass implementation of p-norm."</span>
<span class="k">return</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="o">**</span> <span class="p">(</span><span class="mf">1.</span><span class="o">/</span><span class="n">p</span><span class="p">)</span>
</code></pre></div>
<p>Now, suppose <span class="math">\(|x_i|^p\)</span> causes overflow (for some <span class="math">\(i\)</span>). This will occur for sufficiently large <span class="math">\(p\)</span> or sufficiently large <span class="math">\(x_i\)</span>—even if <span class="math">\(x_i\)</span> is representable (i.e., not NaN or <span class="math">\(\infty\)</span>).</p>
<p>For example:</p>
<div class="highlight"><pre><span></span><code><span class="o">>>></span> <span class="n">big</span> <span class="o">=</span> <span class="mf">1e300</span>
<span class="o">>>></span> <span class="n">x</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="n">big</span><span class="p">])</span>
<span class="o">>>></span> <span class="n">norm1</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="p">[</span> <span class="n">inf</span><span class="p">]</span> <span class="c1"># expected: 1e+300</span>
</code></pre></div>
<p>This fails because we can't square <code>big</code></p>
<div class="highlight"><pre><span></span><code><span class="o">>>></span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">big</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
<span class="p">[</span> <span class="n">inf</span><span class="p">]</span>
</code></pre></div>
<h2>A little math</h2>
<p>There is a way to avoid overflowing because of a few large <span class="math">\(x_i\)</span>.</p>
<p>Here's a little fact about p-norms: for any <span class="math">\(p\)</span> and <span class="math">\(\boldsymbol{x}\)</span>
</p>
<div class="math">$$
|| \alpha \cdot \boldsymbol{x} ||_p = |\alpha| \cdot || \boldsymbol{x} ||_p
$$</div>
<p>We'll use the following version (harder to remember)
</p>
<div class="math">$$
|| \boldsymbol{x} ||_p = |\alpha| \cdot || \boldsymbol{x} / \alpha ||_p
$$</div>
<p>Don't believe it? Here's some algebra:
</p>
<div class="math">$$
\begin{eqnarray*}
|| \boldsymbol{x} ||_p
&=& \left( \sum_i |x_i|^p \right)^{\frac{1}{p}} \\
&=& \left( \sum_i \frac{|\alpha|^p}{|\alpha|^p} \cdot |x_i|^p \right)^{\frac{1}{p}} \\
&=& |\alpha| \cdot \left( \sum_i \left( \frac{|x_i| }{|\alpha|} \right)^p \right)^{\frac{1}{p}} \\
&=& |\alpha| \cdot \left( \sum_i \left| \frac{x_i }{\alpha} \right|^p \right)^{\frac{1}{p}} \\
&=& |\alpha| \cdot || \boldsymbol{x} / \alpha ||_p
\end{eqnarray*}
$$</div>
<h2>Back to numerical stability</h2>
<p>Suppose we pick <span class="math">\(\alpha = \max_i |x_i|\)</span>. Now, the largest number we have to take
the power of is one—making it very difficult to overflow on the account of
<span class="math">\(\boldsymbol{x}\)</span>. This should remind you of the infamous log-sum-exp trick.</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">robust_norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="k">return</span> <span class="n">a</span> <span class="o">*</span> <span class="n">norm1</span><span class="p">(</span><span class="n">x</span> <span class="o">/</span> <span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
</code></pre></div>
<p>Now, our example from before works :-)</p>
<div class="highlight"><pre><span></span><code><span class="o">>>></span> <span class="n">robust_norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="mf">1e+300</span>
</code></pre></div>
<h2>Remarks</h2>
<ul>
<li>
<p>It appears as if <code>scipy.linalg.norm</code> is robust to overflow, while <code>numpy.linalg.norm</code> is not. Note that <code>scipy.linalg.norm</code> appears to be a bit slower.</p>
</li>
<li>
<p>The <code>logsumexp</code> trick is nearly identical, but operates in the log-domain, i.e., <span class="math">\(\text{logsumexp}(\log(|x|) \cdot p) / p = \log || x ||_p\)</span>. You can implement both tricks with the same code, if you use different number classes for log-domain and real-domain—a trick you might have seen before.</p>
</li>
</ul>
<div class="highlight"><pre><span></span><code><span class="kn">from</span> <span class="nn">arsenal.math</span> <span class="kn">import</span> <span class="n">logsumexp</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="nb">abs</span>
<span class="o">>>></span> <span class="n">p</span> <span class="o">=</span> <span class="mi">2</span>
<span class="o">>>></span> <span class="n">x</span> <span class="o">=</span> <span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="o">>>></span> <span class="n">logsumexp</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">*</span> <span class="n">p</span><span class="p">)</span> <span class="o">/</span> <span class="n">p</span>
<span class="mf">1.91432069824</span>
<span class="o">>>></span> <span class="n">log</span><span class="p">(</span><span class="n">robust_norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
<span class="mf">1.91432069824</span>
</code></pre></div>
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</script>KL-divergence as an objective function2014-10-06T00:00:00-04:002014-10-06T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2014-10-06:/blog/post/2014/10/06/kl-divergence-as-an-objective-function/<p>It's well-known that
<a href="http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence">KL-divergence</a>
is not symmetric, but which direction is right for fitting your model?</p>
<h4>Which KL is which? A cheat sheet</h4>
<p>If we're fitting <span class="math">\(q_\theta\)</span> to <span class="math">\(p\)</span> using</p>
<p><span class="math">\(\textbf{KL}(p || q_\theta)\)</span></p>
<ul>
<li>
<p>mean-seeking, <em>inclusive</em> (more principled because approximates the <em>full</em> distribution)</p>
</li>
<li>
<p>requires normalization wrt <span class="math">\(p\)</span> (i.e., often <em>not</em> computationally convenient)</p>
</li>
</ul>
<p><span class="math">\(\textbf{KL}(q_\theta || p)\)</span></p>
<ul>
<li>
<p>mode-seeking, <em>exclusive</em></p>
</li>
<li>
<p>no normalization wrt <span class="math">\(p\)</span> (i.e., computationally convenient)</p>
</li>
</ul>
<p><strong>Mnemonic</strong>: "When the truth comes first, you get the whole truth" (h/t
<a href="https://www.umiacs.umd.edu/~resnik/">Philip Resnik</a>). Here "whole truth"
corresponds to the <em>inclusiveness</em> of <span class="math">\(\textbf{KL}(p || q)\)</span>.</p>
<p>As far as remembering the equation, I pretend that "<span class="math">\(||\)</span>" is a division symbol,
which happens to correspond nicely to a division symbol in the equation (I'm not
sure it's intentional).</p>
<h2>Inclusive vs. exclusive divergence</h2>
<div style="background-color: #f2f2f2; border: 2px solid #ggg; padding: 10px;">
<img src="http://timvieira.github.io/blog/images/KL-inclusive-exclusive.png" />
Figure by <a href="http://www.johnwinn.org/">John Winn</a>.
</div>
<p><br/></p>
<h2>Computational perspecive</h2>
<p>Let's look at what's involved in fitting a model <span class="math">\(q_\theta\)</span> in each
direction. In this section, I'll describe the gradient and pay special attention
to the issue of normalization.</p>
<p><strong>Notation</strong>: <span class="math">\(p,q_\theta\)</span> are probabilty distributions. <span class="math">\(p = \bar{p} / Z_p\)</span>,
where <span class="math">\(Z_p\)</span> is the normalization constant. Similarly for <span class="math">\(q\)</span>.</p>
<h3>The easy direction <span class="math">\(\textbf{KL}(q_\theta || p)\)</span></h3>
<div class="math">\begin{align*}
\textbf{KL}(q_\theta || p)
&= \sum_d q(d) \log \left( \frac{q(d)}{p(d)} \right) \\
&= \sum_d q(d) \left( \log q(d) - \log p(d) \right) \\
&= \underbrace{\sum_d q(d) \log q(d)}_{-\text{entropy}} - \underbrace{\sum_d q(d) \log p(d)}_{\text{cross-entropy}} \\
\end{align*}</div>
<p>Let's look at normalization of <span class="math">\(p\)</span>, the entropy term is easy because there is no <span class="math">\(p\)</span> in it.
</p>
<div class="math">\begin{align*}
\sum_d q(d) \log p(d)
&= \sum_d q(d) \log (\bar{p}(d) / Z_p) \\
&= \sum_d q(d) \left( \log \bar{p}(d) - \log Z_p) \right) \\
&= \sum_d q(d) \log \bar{p}(d) - \sum_d q(d) \log Z_p \\
&= \sum_d q(d) \log \bar{p}(d) - \log Z_p
\end{align*}</div>
<p>In this case, <span class="math">\(-\log Z_p\)</span> is an additive constant, which can be dropped because
we're optimizing.</p>
<p>This leaves us with the following optimization problem:
</p>
<div class="math">\begin{align*}
& \underset{\theta}{\text{argmin}}\, \textbf{KL}(q_\theta || p) \\
&\qquad = \underset{\theta}{\text{argmin}}\, \sum_d q_\theta(d) \log q_\theta(d) - \sum_d q_\theta(d) \log \bar{p}(d)
\end{align*}</div>
<p>Let's work out the gradient
</p>
<div class="math">\begin{align*}
& \nabla\left[ \sum_d q_\theta(d) \log q_\theta(d) - \sum_d q_\theta(d) \log \bar{p}(d) \right] \\
&\qquad = \sum_d \nabla \left[ q_\theta(d) \log q_\theta(d) \right] - \sum_d \nabla\left[ q_\theta(d) \right] \log \bar{p}(d) \\
&\qquad = \sum_d \nabla \left[ q_\theta(d) \right] \left( 1 + \log q_\theta(d) \right) - \sum_d \nabla\left[ q_\theta(d) \right] \log \bar{p}(d) \\
&\qquad = \sum_d \nabla \left[ q_\theta(d) \right] \left( 1 + \log q_\theta(d) - \log \bar{p}(d) \right) \\
&\qquad = \sum_d \nabla \left[ q_\theta(d) \right] \left( \log q_\theta(d) - \log \bar{p}(d) \right) \\
\end{align*}</div>
<p>We killed the one in the last equality because <span class="math">\(\sum_d \nabla
\left[ q(d) \right] = \nabla \left[ \sum_d q(d) \right] = \nabla
\left[ 1 \right] = 0\)</span>, for any <span class="math">\(q\)</span> which is a probability distribution.</p>
<p>This direction is convenient because we don't need to normalize
<span class="math">\(p\)</span>. Unfortunately, the "easy" direction is nonconvex in general—unlike
the "hard" direction, which (as we'll see shortly) is convex.</p>
<h3>Harder direction <span class="math">\(\textbf{KL}(p || q_\theta)\)</span></h3>
<div class="math">\begin{align*}
\textbf{KL}(p || q_\theta)
&= \sum_d p(d) \log \left( \frac{p(d)}{q(d)} \right) \\
&= \sum_d p(d) \left( \log p(d) - \log q(d) \right) \\
&= \sum_d p(d) \log p(d) - \sum_d p(d) \log q(d) \\
\end{align*}</div>
<p>Clearly the first term (entropy) won't matter if we're just trying optimize wrt
<span class="math">\(\theta\)</span>. So, let's focus on the second term (cross-entropy).
</p>
<div class="math">\begin{align*}
\sum_d p(d) \log q(d)
&= \frac{1}{Z_p} \sum_d \bar{p}(d) \log \left( \bar{q}(d)/Z_q \right) \\
&= \frac{1}{Z_p} \sum_d \bar{p}(d) \left( \log \bar{q}(d) - \log Z_q \right) \\
&= \left(\frac{1}{Z_p} \sum_d \bar{p}(d) \log \bar{q}(d)\right) - \left(\frac{1}{Z_p} \sum_d \bar{p}(d) \log Z_q\right) \\
&= \left(\frac{1}{Z_p} \sum_d \bar{p}(d) \log \bar{q}(d)\right) - \left( \log Z_q \right) \left( \frac{1}{Z_p} \sum_d \bar{p}(d)\right) \\
&= \left(\frac{1}{Z_p} \sum_d \bar{p}(d) \log \bar{q}(d)\right) - \log Z_q
\end{align*}</div>
<p>The gradient, when <span class="math">\(q\)</span> is in the exponential family, is intuitive:</p>
<div class="math">\begin{align*}
\nabla \left[ \frac{1}{Z_p} \sum_d \bar{p}(d) \log \bar{q}(d) - \log Z_q \right]
&= \frac{1}{Z_p} \sum_d \bar{p}(d) \nabla \left[ \log \bar{q}(d) \right] - \nabla \log Z_q \\
&= \frac{1}{Z_p} \sum_d \bar{p}(d) \phi_q(d) - \mathbb{E}_q \left[ \phi_q \right] \\
&= \mathbb{E}_p \left[ \phi_q \right] - \mathbb{E}_q \left[ \phi_q \right]
\end{align*}</div>
<p>Why do we say this is hard to compute? Well, for most interesting models, we
can't compute <span class="math">\(Z_p = \sum_d \bar{p}(d)\)</span>. This is because <span class="math">\(p\)</span> is presumed to be a
complex model (e.g., the real world, an intricate factor graph, a complicated
Bayesian posterior). If we can't compute <span class="math">\(Z_p\)</span>, it's highly unlikely that we can
compute another (nontrivial) integral under <span class="math">\(\bar{p}\)</span>, e.g., <span class="math">\(\sum_d \bar{p}(d)
\log \bar{q}(d)\)</span>.</p>
<p>Nonetheless, optimizing KL in this direction is still useful. Examples include:
expectation propagation, variational decoding, and maximum likelihood
estimation. In the case of maximum likelihood estimation, <span class="math">\(p\)</span> is the empirical
distribution, so technically you don't have to compute its normalizing constant,
but you do need samples from it, which can be just as hard to get as computing a
normalization constant.</p>
<p>Optimization problem is <em>convex</em> when <span class="math">\(q_\theta\)</span> is an exponential
family—i.e., for any <span class="math">\(p\)</span> the <em>optimization</em> problem is "easy." You can
think of maximum likelihood estimation (MLE) as a method which minimizes KL
divergence based on samples of <span class="math">\(p\)</span>. In this case, <span class="math">\(p\)</span> is the true data
distribution! The first term in the gradient is based on a sample instead of an
exact estimate (often called "observed feature counts"). The downside, of
course, is that computing <span class="math">\(\mathbb{E}_p \left[ \phi_q \right]\)</span> might not be
tractable or, for MLE, require tons of samples.</p>
<h2>Remarks</h2>
<ul>
<li>
<p>In many ways, optimizing exclusive KL makes no sense at all! Except for the
fact that it's computable when inclusive KL is often not. Exclusive KL is
generally regarded as "an approximation" to inclusive KL. This bias in this
approximation can be quite large.</p>
</li>
<li>
<p>Inclusive vs. exclusive is an important distinction: Inclusive divergences
require <span class="math">\(q > 0\)</span> whenever <span class="math">\(p > 0\)</span> (i.e., no "false negatives"), whereas
exclusive divergences favor a single mode (i.e., only a good fit around a that
mode).</p>
</li>
<li>
<p>When <span class="math">\(q\)</span> is an exponential family, <span class="math">\(\textbf{KL}(p || q_\theta)\)</span> will be convex
in <span class="math">\(\theta\)</span>, no matter how complicated <span class="math">\(p\)</span> is, whereas <span class="math">\(\textbf{KL}(q_\theta
|| p)\)</span> is generally nonconvex (e.g., if <span class="math">\(p\)</span> is multimodal).</p>
</li>
<li>
<p>Computing the value of either KL divergence requires normalization. However,
in the "easy" (exclusive) direction, we can optimize KL without computing
<span class="math">\(Z_p\)</span> (as it results in only an additive constant difference).</p>
</li>
<li>
<p>Both directions of KL are special cases of
<a href="https://en.wikipedia.org/wiki/R%C3%A9nyi_entropy"><span class="math">\(\alpha\)</span>-divergence</a>. For a
unified account of both directions consider looking into <span class="math">\(\alpha\)</span>-divergence.</p>
</li>
</ul>
<h3>Acknowledgments</h3>
<p>I'd like to thank the following people:</p>
<ul>
<li>
<p><a href="https://twitter.com/ryandcotterell">Ryan Cotterell</a> for an email exchange
which spawned this article.</p>
</li>
<li>
<p><a href="https://twitter.com/adveisner">Jason Eisner</a> for teaching me all this stuff.</p>
</li>
<li>
<p><a href="https://twitter.com/florian_shkurti">Florian Shkurti</a> for a useful email
discussion, which caugh a bug in my explanation of why inclusive KL is hard to
compute/optimize.</p>
</li>
<li>
<p><a href="https://twitter.com/sjmielke">Sabrina Mielke</a> for the suggesting the
"inclusive vs. exclusive" figure.</p>
</li>
</ul>
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</script>Complex-step derivative2014-08-07T00:00:00-04:002014-08-07T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2014-08-07:/blog/post/2014/08/07/complex-step-derivative/<p>Estimate derivatives by simply passing in a complex number to your function!</p>
<div class="math">$$
f'(x) \approx \frac{1}{\varepsilon} \text{Im}\Big[ f(x + i \cdot \varepsilon) \Big]
$$</div>
<p><br/>Recall, the centered-difference approximation is a fairly accurate method for
approximating derivatives of a univariate function <span class="math">\(f\)</span>, which only requires two
function evaluations. A similar derivation, based on the Taylor series expansion
with a complex perturbation, gives us a similarly-accurate approximation with a
single (complex) function evaluation instead of two (real-valued) function
evaluations. Note: <span class="math">\(f\)</span> must support complex inputs (in frameworks, such as numpy
or matlab, this often requires no modification to source code).</p>
<p>This post is based on
<a href="http://mdolab.engin.umich.edu/sites/default/files/Martins2003CSD.pdf">Martins+'03</a>.</p>
<p><strong>Derivation</strong>: Start with the Taylor series approximation:</p>
<div class="math">$$
f(x + i \cdot \varepsilon) =
\frac{i^0 \varepsilon^0}{0!} f(x)
+ \frac{i^1 \varepsilon^1}{1!} f'(x)
+ \frac{i^2 \varepsilon^2}{2!} f''(x)
+ \frac{i^3 \varepsilon^3}{3!} f'''(x)
+ \cdots
$$</div>
<p><br/>Take the imaginary part of both sides and solve for <span class="math">\(f'(x)\)</span>. Note: the <span class="math">\(f\)</span> and
<span class="math">\(f''\)</span> term disappear because <span class="math">\(i^0\)</span> and <span class="math">\(i^2\)</span> are real-valued.</p>
<div class="math">$$
f'(x) = \frac{1}{\varepsilon} \text{Im}\Big[ f(x + i \cdot \varepsilon) \Big] + \frac{\varepsilon^2}{3!} f'''(x) + \cdots
$$</div>
<p><br/>As usual, using a small <span class="math">\(\varepsilon\)</span> let's us throw out higher-order
terms. And, we arrive at the following approximation:</p>
<div class="math">$$
f'(x) \approx \frac{1}{\varepsilon} \text{Im}\Big[ f(x + i \cdot \varepsilon) \Big]
$$</div>
<p><br/>If instead, we take the real part and solve for <span class="math">\(f(x)\)</span>, we get an approximation
to the function's value at <span class="math">\(x\)</span>:</p>
<div class="math">$$
f(x) \approx \text{Re}\Big[ f(x + i \cdot \varepsilon) \Big]
$$</div>
<p><br/>In other words, a single (complex) function evaluations computes both the
function's value and the derivative.</p>
<p><strong>Code</strong>:</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">complex_step</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-10</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Higher-order function takes univariate function which computes a value and</span>
<span class="sd"> returns a function which returns value-derivative pair approximation.</span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="nf">f1</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">eps</span><span class="p">))</span> <span class="c1"># convert input to complex number</span>
<span class="k">return</span> <span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">y</span><span class="o">.</span><span class="n">imag</span> <span class="o">/</span> <span class="n">eps</span><span class="p">)</span> <span class="c1"># return function value and gradient</span>
<span class="k">return</span> <span class="n">f1</span>
</code></pre></div>
<p>A simple test:</p>
<div class="highlight"><pre><span></span><code><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="mi">10</span> <span class="c1"># function</span>
<span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c1"># gradient</span>
<span class="n">x</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="nb">print</span> <span class="n">complex_step</span><span class="p">(</span><span class="n">f</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
</code></pre></div>
<p><strong>Other comments</strong></p>
<ul>
<li>
<p>Using the complex-step method to estimate the gradients of multivariate
functions requires independent approximations for each dimension of the
input.</p>
</li>
<li>
<p>Although the complex-step approximation only requires a single function
evaluation, it's unlikely faster than performing two function evaluations
because operations on complex numbers are generally much slower than on floats
or doubles.</p>
</li>
</ul>
<p><strong>Code</strong>: Check out the
<a href="https://gist.github.com/timvieira/3d3db3e5e78e17cdd103">gist</a> for this post.</p>
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</script>Gumbel-max trick and weighted reservoir sampling2014-08-01T00:00:00-04:002014-08-01T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2014-08-01:/blog/post/2014/08/01/gumbel-max-trick-and-weighted-reservoir-sampling/<p>A while back, <a href="http://people.cs.umass.edu/~wallach/">Hanna</a> and I stumbled upon
the following blog post:
<a href="http://blog.cloudera.com/blog/2013/04/hadoop-stratified-randosampling-algorithm">Algorithms Every Data Scientist Should Know: Reservoir Sampling</a>,
which got us excited about reservoir sampling.</p>
<p>Around the same time, I attended a talk by
<a href="http://cs.haifa.ac.il/~tamir/">Tamir Hazan</a> about some of his work on
perturb-and-MAP
<a href="http://cs.haifa.ac.il/~tamir/papers/mean-width-icml12.pdf">(Hazan & Jaakkola, 2012)</a>,
which is inspired by the
<a href="https://hips.seas.harvard.edu/blog/2013/04/06/the-gumbel-max-trick-for-discrete-distributions/">Gumbel-max-trick</a>
(see <a href="/blog/post/2014/07/31/gumbel-max-trick/">previous post</a>). The apparent similarity between weighted reservoir sampling and the Gumbel-max trick lead us to make some cute connections, which I'll describe in this post.</p>
<p><strong>The problem</strong>: We're given a stream of unnormalized probabilities, <span class="math">\(x_1, x_2, \cdots\)</span>. At any point in time <span class="math">\(t\)</span> we'd like to have a sampled index <span class="math">\(i\)</span> available, where the probability of <span class="math">\(i\)</span> is given by <span class="math">\(\pi_t(i) = \frac{x_i}{
\sum_{j=1}^t x_j}\)</span>.</p>
<p>Assume, without loss of generality, that <span class="math">\(x_i > 0\)</span> for all <span class="math">\(i\)</span>. (If any element has a zero weight we can safely ignore it since it should never be sampled.)</p>
<p><strong>Streaming Gumbel-max sampler</strong>: I came up with the following algorithm, which is a simple "modification" of the Gumbel-max-trick for handling streaming data:</p>
<dl>
<dt><span class="math">\(a = -\infty; b = \text{null} \ \ \text{# maximum value and index}\)</span></dt>
<dt>for <span class="math">\(i=1,2,\cdots;\)</span> do:</dt>
<dd># Compute log-unnormalized probabilities</dd>
<dd><span class="math">\(w_i = \log(x_i)\)</span></dd>
<dd># Additively perturb each weight by a Gumbel random variate</dd>
<dd><span class="math">\(z_i \sim \text{Gumbel}(0,1)\)</span></dd>
<dd><span class="math">\(k_i = w_i + z_i\)</span></dd>
<dd># Keep around the largest <span class="math">\(k_i\)</span> (i.e. the argmax)</dd>
<dd>if <span class="math">\(k_i > a\)</span>:</dd>
<dd><span class="math">\(\ \ \ \ a = k_i\)</span></dd>
<dd><span class="math">\(\ \ \ \ b = i\)</span></dd>
</dl>
<p>If we interrupt this algorithm at any point, we have a sample <span class="math">\(b\)</span>.</p>
<p>After convincing myself this algorithm was correct, I sat down to try to
understand the algorithm in the blog post, which is due to Efraimidis and
Spirakis (2005) (<a href="http://dl.acm.org/citation.cfm?id=1138834">paywall</a>,
<a href="http://utopia.duth.gr/~pefraimi/research/data/2007EncOfAlg.pdf">free summary</a>). They
looked similar in many ways but used different sorting keys/perturbations.</p>
<p><strong>Efraimidis and Spirakis (2005)</strong>: Here is the ES algorithm for weighted
reservoir sampling</p>
<dl>
<dt><span class="math">\(a = -\infty; b = \text{null}\)</span></dt>
<dt>for <span class="math">\(i=1,2,\cdots;\)</span> do:</dt>
<dd># compute randomized key</dd>
<dd><span class="math">\(u_i \sim \text{Uniform}(0,1)\)</span></dd>
<dd><span class="math">\(e_i = u_i^{(\frac{1}{x_i})}\)</span></dd>
<dd># Keep around the largest <span class="math">\(e_i\)</span></dd>
<dd>if <span class="math">\(e_i > a\)</span>:</dd>
<dd><span class="math">\(\ \ \ \ a = e_i\)</span></dd>
<dd><span class="math">\(\ \ \ \ b = i\)</span></dd>
</dl>
<p>Again, if we interrupt this algorithm at any point, we have our sample <span class="math">\(b\)</span>. Note
that you can simplify <span class="math">\(e_i\)</span> so that you don't have to compute <code>pow</code> (which is nice
because <code>pow</code> is pretty slow). It's equivalent to use <span class="math">\(e'_i = \log(e_i) =
\log(u_i)/x_i\)</span> because <span class="math">\(\log\)</span> is monotonic. (Note that <span class="math">\(-e'_i \sim
\textrm{Exponential}(x_i)\)</span>.)</p>
<!--
I find this version of the algorithm more intuitive, since it's well-known that
$\left(\underset{{i=1 \ldots t}}{\min} \textrm{Exponential}(x_i) \right) =
\textrm{Exponential}\left(\sum_{i=1}^t x_i \right)$. This version makes it clear
that minimizing is actually summing. However, we want the argmin, which is
distributed according to $\pi_t$.
-->
<p><strong>Relationship</strong>: At a high level, we can see that both algorithms compute a
randomized key and take an argmax. What's the relationship between the keys?</p>
<p>First, note that a <span class="math">\(\text{Gumbel}(0,1)\)</span> variate can be generated via
<span class="math">\(-\log(-\log(\text{Uniform}(0,1)))\)</span>. This is a straightforward application of
the
<a href="http://en.wikipedia.org/wiki/Inverse_transform_sampling">inverse transform sampling</a>
method for random number generation. This means that if we use the same sequence
of uniform random variates then, <span class="math">\(z_i = -\log(-\log(u_i))\)</span>.</p>
<p>However, this does not give use equality between <span class="math">\(k_i\)</span> and <span class="math">\(e_i\)</span>, but it does
turn out that <span class="math">\(k_i = -\log(-\log(e_i))\)</span>, which is useful because this is a
monotonic transformation on the interval <span class="math">\((0,1)\)</span>. Since monotonic
transformations preserve ordering, the sequences <span class="math">\(k\)</span> and <span class="math">\(e\)</span> result in the same
comparison decisions, as well as, the same argmax. In summary, the algorithms
are the same!</p>
<p><strong>Extensions</strong>: After reading a little further along in the ES paper, we see
that the same algorithm can be used to perform <em>sampling without replacement</em> by
sorting and taking the elements with the highest keys. This same modification applies to the Gumbel-max-trick because the keys have precisely the same ordering as ES. In practice, we don't sort the key, but instead, use a bounded priority queue.</p>
<p><strong>Closing</strong>: To the best of my knowledge, the connection between the
Gumbel-max trick and ES is undocumented. Furthermore, the Gumbel-max-trick is not known as a streaming algorithm, much less known to perform sampling without replacement! If you know of a reference, let me know.</p>
<p><strong>How to cite this article</strong>: If you found this article useful, please cite it as</p>
<pre style="background-color: white; color: black; border: #333;">
@misc{vieira2014gumbel,
title = {Gumbel-max trick and weighted reservoir sampling},
author = {Tim Vieira},
url = {http://timvieira.github.io/blog/post/2014/08/01/gumbel-max-trick-and-weighted-reservoir-sampling/},
year = {2014}
}
</pre>
<h2>Further reading</h2>
<p>I have a few other articles that are related to
<a href="http://timvieira.github.io/blog/tag/sampling-without-replacement.html">sampling without replacement</a>,
<a href="http://timvieira.github.io/blog/tag/reservoir-sampling.html">reservoir sampling</a>,
and <a href="http://timvieira.github.io/blog/tag/gumbel.html">Gumbel tricks</a>.</p>
<p>Here are a few interesting papers that build on the ideas in this article.</p>
<ul>
<li>
<p>Wouter Kool, Herke van Hoof, and Max Welling. 2019.
<a href="https://arxiv.org/abs/1903.06059">Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling Sequences Without Replacement</a></p>
</li>
<li>
<p>Sang Michael Xie and Stefano Ermon. 2019. <a href="https://arxiv.org/abs/1901.10517">Reparameterizable Subset Sampling via Continuous Relaxations</a></p>
</li>
</ul>
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</script>Gumbel-max trick2014-07-31T00:00:00-04:002014-07-31T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2014-07-31:/blog/post/2014/07/31/gumbel-max-trick/<p><strong>Goal</strong>: Sampling from a discrete distribution parametrized by unnormalized
log-probabilities:</p>
<div class="math">$$
\pi_k = \frac{1}{z} \exp(x_k) \ \ \ \text{where } z = \sum_{j=1}^K \exp(x_j)
$$</div>
<p><strong>The usual way</strong>: Exponentiate and normalize (using the
<a href="/blog/post/2014/02/11/exp-normalize-trick/">exp-normalize trick</a>), then use the
an algorithm for sampling from a discrete distribution (aka categorical):</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">usual</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">cdf</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">x</span><span class="o">.</span><span class="n">max</span><span class="p">())</span><span class="o">.</span><span class="n">cumsum</span><span class="p">()</span> <span class="c1"># the exp-normalize trick</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">cdf</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cdf</span><span class="o">.</span><span class="n">searchsorted</span><span class="p">(</span><span class="n">u</span> <span class="o">*</span> <span class="n">z</span><span class="p">)</span>
</code></pre></div>
<p><strong>The Gumbel-max trick</strong>:</p>
<div class="math">$$
y = \underset{ i \in \{1,\cdots,K\} }{\operatorname{argmax}} x_i + z_i
$$</div>
<p>where <span class="math">\(z_1 \cdots z_K\)</span> are i.i.d. <span class="math">\(\text{Gumbel}(0,1)\)</span> random variates. It
turns out that <span class="math">\(y\)</span> is distributed according to <span class="math">\(\pi\)</span>. (See the short derivations
in this
<a href="https://lips.cs.princeton.edu/the-gumbel-max-trick-for-discrete-distributions/">blog post</a>.)</p>
<p>Implementing the Gumbel-max trick is remarkable easy:</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">gumbel_max_sample</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">gumbel</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
</code></pre></div>
<p>If you don't have access to a Gumbel random variate generator, you can use
<span class="math">\(-\log(-\log(\text{Uniform}(0,1))\)</span></p>
<p><strong>Comparison</strong>:</p>
<ol>
<li>
<p>Number of calls to the random number generator: Gumbel-max requires <span class="math">\(K\)</span>
samples from a uniform, whereas the usual algorithm only requires <span class="math">\(1\)</span>.</p>
</li>
<li>
<p>Gumbel is a one-pass algorithm: It does not need to see all of the data
(e.g., to normalize) before it can start partially sampling. Thus,
Gumbel-max can be used for
<a href="http://timvieira.github.io/blog/post/2014/08/01/gumbel-max-trick-and-weighted-reservoir-sampling/">weighted sampling from a stream</a>.</p>
</li>
<li>
<p>Low-level efficiency: The Gumbel-max trick requires <span class="math">\(2K\)</span> calls to <span class="math">\(\log\)</span>,
whereas ordinary requires <span class="math">\(K\)</span> calls to <span class="math">\(\exp\)</span>. Since <span class="math">\(\exp\)</span> and <span class="math">\(\log\)</span> are
expensive function, we'd like to avoid calling them. What gives? Well,
Gumbel's calls to <span class="math">\(\log\)</span> do not depend on the data so they can be
precomputed; this is handy for implementations which rely on vectorization
for efficiency, e.g. python+numpy.</p>
</li>
</ol>
<p><strong>Further reading</strong>: I have a few posts relating to the Gumbel-max trick. Have a
look at <a href="/blog/tag/gumbel.html">posts tagged with Gumbel</a>.</p>
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</script>Rant against grid search2014-07-22T00:00:00-04:002014-07-22T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2014-07-22:/blog/post/2014/07/22/rant-against-grid-search/<p>Grid search is a simple and intuitive algorithm for optimizing and/or exploring
the effects of parameters to a function. However, given its rigid definition
grid search is susceptible to degenerate behavior. One type of unfortunate
behavior occurs in the presence of unimportant parameters, which results in many
(potentially expensive) function evaluations being wasted.</p>
<p>This is a very simple point, but nonetheless I'll illustrate with a simple
example.</p>
<p>Consider the following simple example, let's find the argmax of <span class="math">\(f(x,y) = -x^2\)</span>.</p>
<p>Suppose we search over a <span class="math">\(10\)</span>-by-<span class="math">\(10\)</span> grid, resulting in a total of <span class="math">\(100\)</span>
function evaluations. For this function, we expect precision which proportional
of the number of samples in the <span class="math">\(x\)</span>-dimension, which is only <span class="math">\(10\)</span> samples! On
the other hand, randomly sampling points over the same space results in <span class="math">\(100\)</span>
samples in every dimension.</p>
<p>In other words, randomly sample instead of using a rigid grid. If you have
points, which are not uniformly spaced, I'm willing to bet that an appropriate
probability distribution exists.</p>
<p>This type of problem is common on hyperparameter optimizations. For futher
reading see
<a href="http://jmlr.org/papers/volume13/bergstra12a/bergstra12a.pdf">Bergstra & Bengio (2012)</a>.</p>
<p><strong>Other thoughts</strong>:</p>
<ol>
<li>
<p>Local search is often much more effective. For example, gradient-based
optimization, Nelder-Mead, stochastic local search, coordinate ascent.</p>
</li>
<li>
<p>Grid search tends to produce nicer-looking plots.</p>
</li>
<li>
<p>What about variance in the results? Two things: (a) This is a concern for
replicability, but is easily remedied by making sampled parameters
available. (b) There is always some probability that the sampling gives you a
terrible set of points. This shouldn't be a problem if you use enough
samples.</p>
</li>
</ol>
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</script>Expected value of a quadratic and the Delta method2014-07-21T00:00:00-04:002014-07-21T00:00:00-04:00Tim Vieiratag:timvieira.github.io,2014-07-21:/blog/post/2014/07/21/expected-value-of-a-quadratic-and-the-delta-method/<p><strong>Expected value of a quadratic</strong>: Suppose we'd like to compute the expectation of a quadratic function, i.e., <span class="math">\(\mathbb{E}\left[ x^{\top}\negthinspace\negthinspace A x \right]\)</span> , where <span class="math">\(x\)</span> is a random vector and <span class="math">\(A\)</span> is deterministic <em>symmetric</em> matrix. Let <span class="math">\(\mu\)</span> and <span class="math">\(\Sigma\)</span> be the mean and variance of <span class="math">\(x\)</span>. It turns out the expected value of a quadratic has the following simple form:</p>
<div class="math">$$
\mathbb{E}\left[ x^{\top}\negthinspace\negthinspace A x \right]
=
\text{trace}\left( A \Sigma \right) + \mu^{\top}\negthinspace A \mu
$$</div>
<p><strong>Delta Method</strong>: Suppose we'd like to compute expected value of a nonlinear function <span class="math">\(f\)</span> applied our random variable <span class="math">\(x\)</span>,
<span class="math">\(\mathbb{E}\left[ f(x) \right]\)</span>. The Delta method approximates this expection by replacing <span class="math">\(f\)</span> by its second-order Taylor approximation <span class="math">\(\hat{f_{a}}\)</span> taken at some point <span class="math">\(a\)</span></p>
<div class="math">$$
\hat{f_{a}}(x) = f(a) + \nabla f(a)^{\top} (x - a) + \frac{1}{2} (x - a)^\top H(a) (x - a)
$$</div>
<p>The expectation of this Taylor approximation is a quadratic function! Let's try to apply our new equation for the expected value of quadratic. We can use the trick from above with <span class="math">\(A=H(a)\)</span> and <span class="math">\(x = (x-a)\)</span>. Note, the covariance matrix is shift-invariant, and the Hessian is a symmetric matrix!</p>
<div class="math">$$
\begin{aligned}
\mathbb{E}\left[ \hat{f_{a}}(x) \right]
& = \mathbb{E} \left[ f(a) + \nabla\negthinspace f(a)^{\top} (x - a) + \frac{1}{2} (x - a)^{\top} H(a)\, (x - a) \right] \\\
& = f(a) + \nabla\negthinspace f(a)^{\top} ( \mu - a ) + \frac{1}{2} \mathbb{E} \left[ (x - a)^{\top} H(a)\, (x - a) \right] \\\
& = f(a) + \nabla\negthinspace f(a)^{\top} ( \mu - a ) +
\frac{1}{2}\left( \text{trace}\left( H(a) \, \Sigma \right) + (\mu - a)^{\top} H(a)\, (\mu - a) \right)
\end{aligned}
$$</div>
<p>Taking the Taylor expansion around <span class="math">\(\mu\)</span> simplifies the equation as follows</p>
<div class="math">\begin{aligned}
\mathbb{E}\left[ \hat{f_{\mu}} (x) \right]
&= \mathbb{E}\left[ f(\mu) + \nabla\negthinspace f(\mu) (x - \mu) + \frac{1}{2} (x - \mu)^{\top} H(\mu)\, (x - \mu) \right] \\\
&= f(\mu) + \frac{1}{2} \, \text{trace}\Big( H(\mu) \, \Sigma \Big)
\end{aligned}</div>
<p>That looks much more tractable! Error bounds are possible to derive, but outside to scope of this post. For a nice use of the delta method in machine learning see <a href="http://arxiv.org/pdf/1307.1493v2.pdf">(Wager+,'13)</a> and
<a href="http://cs.jhu.edu/~jason/papers/smith+eisner.acl06-risk.pdf">(Smith & Eisner,'06)</a></p>
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</script>Visualizing high-dimensional functions with cross-sections2014-02-12T00:00:00-05:002014-02-12T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2014-02-12:/blog/post/2014/02/12/visualizing-high-dimensional-functions-with-cross-sections/<p>Last September, I gave a talk which included a bunch of two-dimensional plots of
a high-dimensional objective I was developing specialized algorithms for
optimizing. A month later, at least three of my colleagues told me that my plots
had inspired them to make similar plots. The plotting trick is really simple and
not original, but nonetheless I'll still write it up for all to enjoy.</p>
<p><strong>Example plot</strong>: This image shows cross-sections of two related functions: a
non-smooth (black) and a smooth approximating function (blue). The plot shows
that the approximation is faithful to the overall shape, but sometimes
over-smooths. In this case, we miss the maximum, which happens near the middle
of the figure.</p>
<p><img alt="Alt text" src="/blog/images/cross-section.png"></p>
<p><strong>Details</strong>: Let <span class="math">\(f: \mathbb{R}^d \rightarrow \mathbb{R}\)</span> be a high-dimensional
function (<span class="math">\(d \gg 2\)</span>), which you'd like to visualize. Unfortunately, you are like
me and can't see in high-dimensions what do you do?</p>
<p>One simple thing to do is take a nonzero vector <span class="math">\(\boldsymbol{d} \in
\mathbb{R}^d\)</span>, take a point of interest <span class="math">\(\boldsymbol{x}\)</span>, and build a local
picture of <span class="math">\(f\)</span> by evaluating it at various intervals along the chosen direction
as follows,</p>
<div class="math">$$
f_i = f(\boldsymbol{x} + \alpha_i \ \boldsymbol{d}) \ \ \text{for } \alpha_i \in [\alpha_\min, \alpha_\max]
$$</div>
<p>Of course, you'll have to pick a reasonable range and discretize it. Note,
<span class="math">\(\boldsymbol{x}\)</span> and <span class="math">\(\boldsymbol{d}\)</span> are fixed for all <span class="math">\(\alpha_i\)</span>. Now, you can
plot <span class="math">\((\alpha_i,f_i)\)</span>.</p>
<p><strong>Picking directions</strong>: There are many alternatives for picking
<span class="math">\(\boldsymbol{d}\)</span>, my favorites are:</p>
<ol>
<li>
<p>Coordinate vectors: Varying one (or two) dimensions.</p>
</li>
<li>
<p>Gradient (if it exists), this direction is guaranteed to show a local
increase/decrease in the objective, unless it's zero because we're at a
local optimum. Some variations on "descent" directions:</p>
<ul>
<li>
<p>Use the gradient direction of a <em>different</em> objective, e.g., plot
(nondifferentiable) accuracy on dev data along the (differentiable)
likelihood direction on training data.</p>
</li>
<li>
<p>Optimizer trajectory: Use PCA on the optimizer's trajectory to find the
directions which summarize the most variation.</p>
</li>
</ul>
</li>
<li>
<p>The difference of two interesting points, e.g., the start and end points of
your optimization, two different solutions.</p>
</li>
<li>
<p>Random:</p>
<p>If all your parameters are on an equal scale, I recommend directions drawn
from a spherical Gaussian.<sup id="sf-visualizing-high-dimensional-functions-with-cross-sections-1-back"><a href="#sf-visualizing-high-dimensional-functions-with-cross-sections-1" class="simple-footnote" title="More formally, vectors drawn from a spherical Gaussian are points uniformly distributed on the surface of a \(d\)-dimensional unit sphere, \(\mathbb{S}^d\). Sampling a vector from a spherical Gaussian is straightforward: sample \(\boldsymbol{d'} \sim \mathcal{N}(\boldsymbol{0},\boldsymbol{I})\), \(\boldsymbol{d} = \boldsymbol{d'} / \| \boldsymbol{d'} \|_2\)">1</a></sup>
The reason being that such a
vector is uniformly distributed across all unit-length directions (i.e., the
angle of the vector, not it's length). We will vary the length ourselves via
<span class="math">\(\alpha\)</span>.</p>
<p>However, often components of <span class="math">\(\boldsymbol{x}\)</span> have different scales, so
finding a "natural scale" is crucial if we are going to draw conclusions
that require a comparison of the perturbation sensitivities across several
dimensions—this is closely related to why we like second-order and
adaptive optimization algorithms
(<a href="https://timvieira.github.io/blog/post/2016/05/27/dimensional-analysis-of-gradient-ascent/">discussion</a>);
<span class="math">\(\boldsymbol{d}\)</span>'s units must match the units of <span class="math">\(\boldsymbol{x}\)</span> in each
coordinate!</p>
</li>
<li>
<p>Maximize "interestingness": You can also use a direction-optimization
procedure to maximize some measure of "interestingness" (e.g., the direction
in which training and dev loss differ the most; the "bumpiest" direction or
direction taking the biggest range of values).</p>
</li>
</ol>
<p><strong>Extension to 3d</strong>: It's pretty easy to extend these ideas to generating
three-dimensional plots by using two vectors, <span class="math">\(\boldsymbol{d_1}\)</span> and
<span class="math">\(\boldsymbol{d_2},\)</span> and varying two parameters <span class="math">\(\alpha\)</span> and <span class="math">\(\beta\)</span>,</p>
<div class="math">$$
f(\boldsymbol{x} + \alpha \ \boldsymbol{d_1} + \beta \ \boldsymbol{d_2})
$$</div>
<p><strong>Closing remarks</strong>: These types of plots are probably best used to: empirically
verify/explore properties of an objective function, compare approximations, test
sensitivity to certain parameters/hyperparameters, visually debug optimization
algorithms.</p>
<p><strong>Further reading</strong>:</p>
<ul>
<li><a href="https://arxiv.org/abs/1712.09913">Visualizing the Loss Landscape of Neural Nets</a></li>
</ul>
<h2>Footnotes</h2>
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</script><ol class="simple-footnotes"><li id="sf-visualizing-high-dimensional-functions-with-cross-sections-1">More formally, vectors drawn from a spherical Gaussian are
points uniformly distributed on the surface of a <span class="math">\(d\)</span>-dimensional unit sphere,
<span class="math">\(\mathbb{S}^d\)</span>. Sampling a vector from a spherical Gaussian is straightforward:
sample <span class="math">\(\boldsymbol{d'} \sim \mathcal{N}(\boldsymbol{0},\boldsymbol{I})\)</span>,
<span class="math">\(\boldsymbol{d} = \boldsymbol{d'} / \| \boldsymbol{d'} \|_2\)</span> <a href="#sf-visualizing-high-dimensional-functions-with-cross-sections-1-back" class="simple-footnote-back">↩</a></li></ol>Exp-normalize trick2014-02-11T00:00:00-05:002014-02-11T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2014-02-11:/blog/post/2014/02/11/exp-normalize-trick/<p>This trick is the very close cousin of the infamous log-sum-exp trick
(<a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.misc.logsumexp.html">scipy.misc.logsumexp</a>).</p>
<p>Supposed you'd like to evaluate a probability distribution <span class="math">\(\boldsymbol{\pi}\)</span>
parametrized by a vector <span class="math">\(\boldsymbol{x} \in \mathbb{R}^n\)</span> as follows:</p>
<div class="math">$$
\pi_i = \frac{ \exp(x_i) }{ \sum_{j=1}^n \exp(x_j) }
$$</div>
<p>The exp-normalize trick leverages the following identity to avoid numerical
overflow. For any <span class="math">\(b \in \mathbb{R}\)</span>,</p>
<div class="math">$$
\pi_i
= \frac{ \exp(x_i - b) \exp(b) }{ \sum_{j=1}^n \exp(x_j - b) \exp(b) }
= \frac{ \exp(x_i - b) }{ \sum_{j=1}^n \exp(x_j - b) }
$$</div>
<p>In other words, the <span class="math">\(\boldsymbol{\pi}\)</span> is shift-invariant. A reasonable choice
is <span class="math">\(b = \max_{i=1}^n x_i\)</span>. With this choice, overflow due to <span class="math">\(\exp\)</span> is
impossible<span class="math">\(-\)</span>the largest number exponentiated after shifting is <span class="math">\(0\)</span>.</p>
<p>The naive implementation is terrible when there are large numbers!</p>
<div class="highlight"><pre><span></span><code><span class="o">>>></span> <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1000</span><span class="p">])</span>
<span class="o">>>></span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="ne">RuntimeWarning</span><span class="p">:</span> <span class="n">overflow</span> <span class="n">encountered</span> <span class="ow">in</span> <span class="n">exp</span>
<span class="ne">RuntimeWarning</span><span class="p">:</span> <span class="n">invalid</span> <span class="n">value</span> <span class="n">encountered</span> <span class="ow">in</span> <span class="n">true_divide</span>
<span class="n">Out</span><span class="p">[</span><span class="mi">4</span><span class="p">]:</span> <span class="n">array</span><span class="p">([</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="n">nan</span><span class="p">])</span>
</code></pre></div>
<p>The exp-normalize trick avoids this common problem.</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">exp_normalize</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
<span class="k">return</span> <span class="n">y</span> <span class="o">/</span> <span class="n">y</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="o">>>></span> <span class="n">exp_normalize</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">array</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">])</span>
</code></pre></div>
<p><strong>Log-sum-exp for computing the log-distibution</strong></p>
<div class="math">$$
\log \pi_i = x_i - \mathrm{logsumexp}(\boldsymbol{x})
$$</div>
<p>where
</p>
<div class="math">$$
\mathrm{logsumexp}(\boldsymbol{x}) = b + \log \sum_{j=1}^n \exp(x_j - b)
$$</div>
<p>Typically with the same choice for <span class="math">\(b\)</span> as above.</p>
<p><strong>Exp-normalize v. log-sum-exp</strong></p>
<p>Exp-normalize is the gradient of log-sum-exp. So you probably need to know both
tricks!</p>
<p>If what you want to remain in log-space, that is, compute
<span class="math">\(\log(\boldsymbol{\pi})\)</span>, you should use logsumexp. However, if
<span class="math">\(\boldsymbol{\pi}\)</span> is your goal, then exp-normalize trick is for you! Since it
avoids additional calls to <span class="math">\(\exp\)</span>, which would be required if using log-sum-exp
and more importantly exp-normalize is more numerically stable!</p>
<p><strong>Numerically stable sigmoid function</strong></p>
<p>The sigmoid function can be computed with the exp-normalize trick in order to
avoid numerical overflow. In the case of <span class="math">\(\text{sigmoid}(x)\)</span>, we have a
distribution with unnormalized log probabilities <span class="math">\([x,0]\)</span>, where we are only
interested in the probability of the first event. From the exp-normalize
identity, we know that the distributions <span class="math">\([x,0]\)</span> and <span class="math">\([0,-x]\)</span> are equivalent (to
see why, plug in <span class="math">\(b=\max(0,x)\)</span>). This is why sigmoid is often expressed in one
of two equivalent ways:</p>
<div class="math">$$
\text{sigmoid}(x) = 1/(1+\exp(-x)) = \exp(x) / (\exp(x) + 1)
$$</div>
<p>Interestingly, each version covers an extreme case: <span class="math">\(x=\infty\)</span> and <span class="math">\(x=-\infty\)</span>,
respectively. Below is some python code which implements the trick:</p>
<div class="highlight"><pre><span></span><code><span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="s2">"Numerically stable sigmoid function."</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">>=</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span>
<span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># if x is less than zero then z will be small, denom can't be</span>
<span class="c1"># zero because it's 1+z.</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">return</span> <span class="n">z</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span>
</code></pre></div>
<p><strong>Closing remarks:</strong> The exp-normalize distribution is also known as a
<a href="https://en.wikipedia.org/wiki/Gibbs_measure">Gibbs measure</a> (sometimes called a
Boltzmann distribution) when it is augmented with a temperature
parameter. Exp-normalize is often called "softmax," which is unfortunate because
log-sum-exp is <em>also</em> called "softmax." However, unlike exp-normalize, it
<em>earned</em> the name because it is acutally a soft version of the max function,
where as exp-normalize is closer to "soft argmax." Nonetheless, most people
still call exp-normalize "softmax."</p>
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</script>Gradient-vector product2014-02-10T00:00:00-05:002014-02-10T00:00:00-05:00Tim Vieiratag:timvieira.github.io,2014-02-10:/blog/post/2014/02/10/gradient-vector-product/<p>We've all written the following test for our gradient code (known as the
finite-difference approximation).</p>
<div class="math">$$
\frac{\partial}{\partial x_i} f(\boldsymbol{x}) \approx
\frac{1}{2 \varepsilon} \Big(
f(\boldsymbol{x} + \varepsilon \cdot \boldsymbol{e_i})
- f(\boldsymbol{x} - \varepsilon \cdot \boldsymbol{e_i})
\Big)
$$</div>
<p>where <span class="math">\(\varepsilon > 0\)</span> and <span class="math">\(\boldsymbol{e_i}\)</span> is a vector of zeros except at
<span class="math">\(i\)</span> where it is <span class="math">\(1\)</span>. This approximation is exact in the limit, and accurate to
<span class="math">\(o(\varepsilon)\)</span> additive error.</p>
<p>This is a specific instance of a more general approximation! The dot product of
the gradient and any (conformable) vector <span class="math">\(\boldsymbol{d}\)</span> can be approximated
with the following formula,</p>
<div class="math">$$
\nabla f(\boldsymbol{x})^{\top} \boldsymbol{d} \approx
\frac{1}{2 \varepsilon} \Big(
f(\boldsymbol{x} + \varepsilon \cdot \boldsymbol{d})
- f(\boldsymbol{x} - \varepsilon \cdot \boldsymbol{d})
\Big)
$$</div>
<p>We get the special case above when <span class="math">\(\boldsymbol{d}=\boldsymbol{e_i}\)</span>. This also
exact in the limit and just as accurate.</p>
<p><strong>Runtime?</strong> Finite-difference approximation is probably too slow for
approximating a high-dimensional gradient because the number of function
evaluations required is <span class="math">\(2 n\)</span> where <span class="math">\(n\)</span> is the dimensionality of <span class="math">\(x\)</span>. However,
if the end goal is to approximate a gradient-vector product, a mere <span class="math">\(2\)</span>
function evaluations is probably faster than specialized code for computing
the gradient.</p>
<p><strong>How to set <span class="math">\(\varepsilon\)</span>?</strong> The second approach is more sensitive to
<span class="math">\(\varepsilon\)</span> because <span class="math">\(\boldsymbol{d}\)</span> is arbitrary, unlike
<span class="math">\(\boldsymbol{e_i}\)</span>, which is a simple unit-norm vector. Luckily some guidance
is available. Andrei (2009) reccommends</p>
<div class="math">$$
\varepsilon = \sqrt{\epsilon_{\text{mach}}} (1 + \|\boldsymbol{x} \|_{\infty}) / \| \boldsymbol{d} \|_{\infty}
$$</div>
<p>where <span class="math">\(\epsilon_{\text{mach}}\)</span> is
<a href="http://en.wikipedia.org/wiki/Machine_epsilon">machine epsilon</a>. (Numpy users:
<code>numpy.finfo(x.dtype).eps</code>).</p>
<h2>Why do I care?</h2>
<ol>
<li>
<p>Well, I tend to work on sparse, but high-dimensional problems where
finite-difference would be too slow. Thus, my usual solution is to only test
several randomly selected dimensions<span class="math">\(-\)</span>biasing samples toward dimensions
which should be nonzero. With the new trick, I can effectively test more
dimensions at once by taking random vectors <span class="math">\(\boldsymbol{d}\)</span>. I recommend
sampling <span class="math">\(\boldsymbol{d}\)</span> from a spherical Gaussian so that we're uniform on
the angle of the vector.</p>
</li>
<li>
<p>Sometimes the gradient-vector dot product is the end goal. This is the case
with Hessian-vector products, which arises in many optimization algorithms,
such as stochastic meta descent. Hessian-vector products are an instance of
the gradient-vector dot product because the Hessian is just the gradient of
the gradient.</p>
</li>
</ol>
<h2>Hessian-vector product</h2>
<p>Hessian-vector products are an instance of the gradient-vector dot product
because since the Hessian is just the gradient of the gradient! Now you only
need to remember one formula!</p>
<div class="math">$$
H(\boldsymbol{x})\, \boldsymbol{d} \approx
\frac{1}{2 \varepsilon} \Big(
\nabla f(\boldsymbol{x} + \varepsilon \cdot \boldsymbol{d})
- \nabla f(\boldsymbol{x} - \varepsilon \cdot \boldsymbol{d})
\Big)
$$</div>
<p>With this trick you never have to actually compute the gnarly Hessian! More on
<a href="http://justindomke.wordpress.com/2009/01/17/hessian-vector-products/">Justin Domke's blog</a></p>
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