Generalization at the Edge of Stability

TL;DR

This paper investigates how training neural networks at the edge of stability, characterized by chaotic dynamics, affects generalization, introducing a new dimension-based theory supported by experiments.

Contribution

It introduces the sharpness dimension, a novel measure linking chaotic optimization dynamics to generalization, supported by theoretical bounds and empirical validation.

Findings

Generalization depends on the full Hessian spectrum and partial determinants.

Chaotic regimes can improve generalization performance.

The theory provides insights into the grokking phenomenon.

Abstract

Training modern neural networks often relies on large learning rates, operating at the edge of stability, where the optimization dynamics exhibit oscillatory and chaotic behavior. Empirically, this regime often yields improved generalization performance, yet the underlying mechanism remains poorly understood. In this work, we represent stochastic optimizers as random dynamical systems, which often converge to a fractal attractor set (rather than a point) with a smaller intrinsic dimension. Building on this connection and inspired by Lyapunov dimension theory, we introduce a novel notion of dimension, coined the `sharpness dimension', and prove a generalization bound based on this dimension. Our results show that generalization in the chaotic regime depends on the complete Hessian spectrum and the structure of its partial determinants, highlighting a complexity that cannot be captured by the trace or spectral norm considered in prior work. Experiments across various MLPs and transformers validate our theory while also providing new insights into the recently observed phenomenon of grokking.