The Anisotropic Noise in Stochastic Gradient Descent: Its Behavior of Escaping from Sharp Minima and Regularization Effects

TL;DR

This paper analyzes how anisotropic noise in SGD helps escape sharp minima and acts as a regularizer, leading to better generalization in deep neural networks.

Contribution

It introduces a novel indicator for noise efficiency, establishes conditions favoring anisotropic noise, and demonstrates its benefits over isotropic noise through systematic experiments.

Findings

Anisotropic noise effectively escapes sharp minima.

Conditions favoring anisotropic over isotropic noise are established.

Experiments confirm improved generalization with anisotropic noise.

Abstract

Understanding the behavior of stochastic gradient descent (SGD) in the context of deep neural networks has raised lots of concerns recently. Along this line, we study a general form of gradient based optimization dynamics with unbiased noise, which unifies SGD and standard Langevin dynamics. Through investigating this general optimization dynamics, we analyze the behavior of SGD on escaping from minima and its regularization effects. A novel indicator is derived to characterize the efficiency of escaping from minima through measuring the alignment of noise covariance and the curvature of loss function. Based on this indicator, two conditions are established to show which type of noise structure is superior to isotropic noise in term of escaping efficiency. We further show that the anisotropic noise in SGD satisfies the two conditions, and thus helps to escape from sharp and poor minima effectively, towards more stable and flat minima that typically generalize well. We systematically design various experiments to verify the benefits of the anisotropic noise, compared with full gradient descent plus isotropic diffusion (i.e. Langevin dynamics).