Graduate Descent

Dimensional analysis of gradient ascent

optimization calculus

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.

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.

Gradient ascent is an iterative procedure for (locally) maximizing a function, \(f: \mathbb{R}^d \mapsto \mathbb{R}\).

$$ x_{t+1} = x_t + \alpha \frac{\partial f(x_t)}{\partial x} $$

In general, \(\alpha\) is a \(d \times d\) matrix, but often we constrain the matrix to be simple, e.g., \(a\cdot I\) for some scalar \(a\) or \(\text{diag}(a)\) for some vector \(a\).

Now, let's look at the units of the change in \(\Delta x=x_{t+1} - x_t\),

$$ (\textbf{units }\Delta x) = \left(\textbf{units }\alpha\cdot \frac{\partial f(x_t)}{\partial x}\right) = (\textbf{units }\alpha) \frac{(\textbf{units }f)}{(\textbf{units }x)}. $$

The units of \(\Delta x\) must be \((\textbf{units }x)\). However, if we assume \(f\) is unit free, we're happy with \((\textbf{units }x) / (\textbf{units }f)\).

Solving for the units of \(\alpha\) we get,

$$ (\textbf{units }\alpha) = \frac{(\textbf{units }x)^2}{(\textbf{units }f)}. $$

This gives us an idea for what \(\alpha\) should be.

For example, the inverse Hessian passes the unit check (if we assume \(f\) 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 \(> 0\)).

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.

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 \(x\)). 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 \(t\).

  • AdaGrad doesn't doesn't pass the unit check. This motivated AdaDelta.

  • AdaDelta uses the ratio of (running estimates of) the root-mean-squares of \(\Delta x\) and \(\partial f / \partial x\). The mean is taken using an exponentially weighted moving average. See paper for actual implementation.

  • Adam came later and made some tweaks to remove (unintended) bias in the AdaDelta estimates of the numerator and denominator.

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.