High dimensional analysis reveals conservative sharpening and a stochastic edge of stability

TL;DR

This paper investigates how stochasticity affects the dynamics of large eigenvalues in the training loss Hessian, revealing conservative sharpening and a stochastic edge of stability, with theoretical insights and experimental validation.

Contribution

It provides a theoretical explanation for the slowdown in eigenvalue growth under stochastic training and introduces a stochastic edge of stability influenced by the Neural Tangent Kernel.

Findings

Eigenvalue growth slows in stochastic training compared to full batch.

A stochastic edge of stability depends on the Neural Tangent Kernel trace.

Controlling stochastic stability can improve optimization performance.

Abstract

Recent empirical and theoretical work has shown that the dynamics of the large eigenvalues of the training loss Hessian have some remarkably robust features across models and datasets in the full batch regime. There is often an early period of progressive sharpening where the large eigenvalues increase, followed by stabilization at a predictable value known as the edge of stability. Previous work showed that in the stochastic setting, the eigenvalues increase more slowly - a phenomenon we call conservative sharpening. We provide a theoretical analysis of a simple high-dimensional model which shows the origin of this slowdown. We also show that there is an alternative stochastic edge of stability which arises at small batch size that is sensitive to the trace of the Neural Tangent Kernel rather than the large Hessian eigenvalues. We conduct an experimental study which highlights the qualitative differences from the full batch phenomenology, and suggests that controlling the stochastic edge of stability can help optimization.