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

TLDR

SGD behavior in deep neural networks has recently attracted significant attention. The study investigates a general unbiased‑noise gradient dynamics that unifies SGD and Langevin dynamics to analyze SGD's ability to escape minima and its regularization effects. The authors derive an indicator measuring alignment between noise covariance and loss curvature, establish two conditions favoring anisotropic noise over isotropic noise for escaping efficiency, and design experiments comparing anisotropic SGD to full gradient descent plus isotropic diffusion. Anisotropic noise in SGD satisfies these conditions, enabling efficient escape from sharp minima toward flatter minima that generalize better.

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).