Dynamical mean-field theory for stochastic gradient descent in Gaussian mixture classification

TL;DR

This paper applies dynamical mean-field theory to analyze the learning dynamics of stochastic gradient descent in high-dimensional Gaussian mixture classification, revealing insights into its performance and landscape navigation.

Contribution

It introduces a dynamical mean-field framework for understanding SGD in non-convex Gaussian mixture classification tasks, extending to a continuous-time stochastic gradient flow.

Findings

SGD dynamics can be characterized by a self-consistent stochastic process.

The analysis reveals how control parameters influence algorithm performance.

The framework bridges high-dimensional statistical physics and neural network training.

Abstract

We analyze in a closed form the learning dynamics of stochastic gradient descent (SGD) for a single-layer neural network classifying a high-dimensional Gaussian mixture where each cluster is assigned one of two labels. This problem provides a prototype of a non-convex loss landscape with interpolating regimes and a large generalization gap. We define a particular stochastic process for which SGD can be extended to a continuous-time limit that we call stochastic gradient flow. In the full-batch limit, we recover the standard gradient flow. We apply dynamical mean-field theory from statistical physics to track the dynamics of the algorithm in the high-dimensional limit via a self-consistent stochastic process. We explore the performance of the algorithm as a function of the control parameters shedding light on how it navigates the loss landscape.