On the diffusion approximation of nonconvex stochastic gradient descent

TL;DR

This paper rigorously analyzes how stochastic gradient descent (SGD) in nonconvex optimization can be approximated by diffusion processes, revealing insights into its escape dynamics from local minima and saddle points, and the influence of batch size.

Contribution

It provides a rigorous diffusion approximation of SGD in nonconvex problems and explores how batch size affects escape from minima and saddle points.

Findings

Diffusion process approximates SGD weakly in small step size regime.

SGD escapes local minima exponentially and saddle points linearly depending on inverse step size.

Small batch sizes help SGD escape unstable points and sharp minima, suggesting larger batch sizes for better generalization.

Abstract

We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small step size regime and the presence of omnidirectional noise, our weak approximating diffusion process suggests the following dynamics for the SGD iteration starting from a local minimizer (resp.~saddle point): it escapes in a number of iterations exponentially (resp.~almost linearly) dependent on the inverse stepsize. The results are obtained using the theory for random perturbations of dynamical systems (theory of large deviations for local minimizers and theory of exiting for unstable stationary points). In addition, we discuss the effects of batch size for the deep neural networks, and we find that small batch size is helpful for SGD algorithms to escape unstable stationary points and sharp minimizers. Our theory indicates that one should increase the batch size at later stage for the SGD to be trapped in flat minimizers for better generalization.