Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks

TL;DR

This paper uncovers how stochastic gradient descent (SGD) implicitly biases neural networks towards simpler subnetworks by attracting parameters to invariant sets, which enhances generalization and explains benefits of large learning rates.

Contribution

It introduces the concept of invariant sets and stochastic attractivity in SGD, revealing a bias towards simpler subnetworks and providing a mechanistic explanation for improved generalization.

Findings

SGD exhibits stochastic attractivity towards invariant sets.

Increased noise levels strengthen the attraction to simpler subnetworks.

Simplification via stochastic collapse improves generalization in neural networks.

Abstract

In this work, we reveal a strong implicit bias of stochastic gradient descent (SGD) that drives overly expressive networks to much simpler subnetworks, thereby dramatically reducing the number of independent parameters, and improving generalization. To reveal this bias, we identify invariant sets, or subsets of parameter space that remain unmodified by SGD. We focus on two classes of invariant sets that correspond to simpler (sparse or low-rank) subnetworks and commonly appear in modern architectures. Our analysis uncovers that SGD exhibits a property of stochastic attractivity towards these simpler invariant sets. We establish a sufficient condition for stochastic attractivity based on a competition between the loss landscape's curvature around the invariant set and the noise introduced by stochastic gradients. Remarkably, we find that an increased level of noise strengthens attractivity, leading to the emergence of attractive invariant sets associated with saddle-points or local maxima of the train loss. We observe empirically the existence of attractive invariant sets in trained deep neural networks, implying that SGD dynamics often collapses to simple subnetworks with either vanishing or redundant neurons. We further demonstrate how this simplifying process of stochastic collapse benefits generalization in a linear teacher-student framework. Finally, through this analysis, we mechanistically explain why early training with large learning rates for extended periods benefits subsequent generalization.