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

Feng Chen, Daniel Kunin, Atsushi Yamamura, Surya Ganguli

Keywords: Implicit Bias, SGD Dynamics, Implicit regularization, Learning rate schedule, Stochastic Gradient Descent, Invariant set, Attractive saddle points, Stochastic collapse, Permutation invariance, Simplicity bias, Teacher-student

TL;DR: This study reveals how stochasticity from mini-batch gradients biases overparameterized neural networks towards "invariant sets" corresponding to simpler subnetworks with improved generalization.

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.

Submission Number: 9367