On the Heavy-Tailed Theory of Stochastic Gradient Descent for Deep Neural Networks

TL;DR

This paper challenges the Gaussian noise assumption in SGD analysis for deep learning, proposing a heavy-tailed -stable distribution model driven by Lévy motion, which better explains the convergence and minima transition behaviors.

Contribution

It introduces a heavy-tailed -stable noise model for SGD, linking tail index to convergence rates and providing experimental validation across various deep learning settings.

Findings

SGD noise is highly non-Gaussian with heavy tails

Heavy-tailed noise influences transition from narrow to wide minima

Convergence rate is explicitly connected to the tail index 

Abstract

The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the \emph{classical} central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic differential equation (SDE) driven by a Brownian motion. We argue that the Gaussianity assumption might fail to hold in deep learning settings and hence render the Brownian motion-based analyses inappropriate. Inspired by non-Gaussian natural phenomena, we consider the GN in a more general context and invoke the \emph{generalized} CLT, which suggests that the GN converges to a \emph{heavy-tailed} $\alpha$-stable random vector, where \emph{tail-index} $\alpha$ determines the heavy-tailedness of the distribution. Accordingly, we propose to analyze SGD as a discretization of an SDE driven by a L\'{e}vy motion. Such SDEs can incur `jumps', which force the SDE and its discretization \emph{transition} from narrow minima to wider minima, as proven by existing metastability theory and the extensions that we proved recently. In this study, under the $\alpha$-stable GN assumption, we further establish an explicit connection between the convergence rate of SGD to a local minimum and the tail-index $\alpha$. To validate the $\alpha$-stable assumption, we conduct experiments on common deep learning scenarios and show that in all settings, the GN is highly non-Gaussian and admits heavy-tails. We investigate the tail behavior in varying network architectures and sizes, loss functions, and datasets. Our results open up a different perspective and shed more light on the belief that SGD prefers wide minima.