# Why control a continuous approximation of stochastic gradient descent instead of just the SGD?

In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in

$$x_{k+1} = x_k - \eta u_k \nabla f_{\gamma_k}(x_k),$$

by an SDE, a so called stochastic modified equation (SME)

$$d X_t = -u_t \nabla f(X_t)dt + u_t \sqrt{\eta \Sigma(X_t)}dW_t,$$

in the sense of weak approximation of order 1 by viewing the SGD update as the application of the Euler scheme to the SDE.

Here

• $$\eta > 0$$ is the maximum learning rate
• $$T > 0$$
• $$u : [0,T] \to [0,1]$$ controls the learning rate
• $$(\gamma_k)_k$$ are iid RV with values in $$\{1,\dots, n\}$$ and $$\mathbb P(\gamma_k = i) = 1/n$$, which represent random samples from a training set (e.g.)
• $$f(x) = \frac 1 n \sum_{i=1}^n f_i(x)$$ is the objective
• $$(W_t)_t$$ is Brownian motion independent of the $$\gamma_k$$

and

$$\Sigma(x) := \frac 1 n \sum_{i=1}^n (\nabla f(x) - \nabla f_i(x))(\nabla f(x) - \nabla f_i(x))^T.$$

Of course, many conditions are left implicit here, s.t. the $$f_i$$ are continuously differentiable or that the coefficients of the SME satisfy growth conditions that guarantee the existence of a unique solution.

Now, the goal is to solve a control problem for the SME, such as

$$\min_{u} \mathbb E(f(X_T)),$$

and use the solution to control the learning rate of the original SGD. What I wonder is

Why can't we just control the original SGD directly? What can we gain by passing to this continuous problem?

Among other things, I think this is important because the difference between the real $$x_k$$ and its continuous counterpart ("$$|X_{\eta k} - x_k|$$") can be quite large since the approximation is only weak. I think this prevents us from letting $$u$$ be dependent on $$X_t$$ as well, because then the solution cannot be meaningfully used for the original SGD.

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