Random Matrix Theory for Stochastic Gradient Descent

TL;DR

This paper applies random matrix theory and Dyson Brownian motion to analyze stochastic gradient descent, deriving a linear scaling rule and identifying universal properties of weight matrix dynamics in neural networks.

Contribution

It introduces a novel application of random matrix theory to describe SGD dynamics, deriving a linear scaling rule and distinguishing universal from non-universal behaviors.

Findings

Derived the linear scaling rule between learning rate and batch size.

Identified universal aspects of weight matrix dynamics.

Validated findings in Gaussian RBM and linear neural networks.

Abstract

Investigating the dynamics of learning in machine learning algorithms is of paramount importance for understanding how and why an approach may be successful. The tools of physics and statistics provide a robust setting for such investigations. Here we apply concepts from random matrix theory to describe stochastic weight matrix dynamics, using the framework of Dyson Brownian motion. We derive the linear scaling rule between the learning rate (step size) and the batch size, and identify universal and non-universal aspects of weight matrix dynamics. We test our findings in the (near-)solvable case of the Gaussian Restricted Boltzmann Machine and in a linear one-hidden-layer neural network.