SGD at the Edge of Stability: Stochastic Stabilization with Large Learning Rates

arXiv:2606.30930 · stat.ML, cs.LG, math.OC · Submitted 2026-06-29 · Read on arXiv

Konstantinos Emmanouilidis, Lachlan MacDonald, Salma Tarmoun, Rene Vidal

stat.ML, cs.LG, math.OC

Submitted: 2026-06-29

License: http://creativecommons.org/licenses/by/4.0/

The gist: Modern deep learning has been shown to operate at the edge of stability, routinely using learning rates far larger than those justified by classical optimization theory.

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Abstract

Modern deep learning has been shown to operate at the edge of stability, routinely using learning rates far larger than those justified by classical optimization theory. Most prior analyses of the edge of stability phenomenon focus on deterministic gradient descent, leaving the stochastic setting largely unexplored. In this work, we provide sharp convergence guarantees for Stochastic Gradient Descent (SGD) applied to the multiclass cross-entropy loss, for both linear classifiers and two-layer neural networks. We show that the stochasticity of SGD may cause the dynamics to alternate between an edge-of-stability regime that is dominated by curvature-driven oscillations, and a stable regime in which the expected loss decreases at a controlled rate. Despite that, we prove that SGD self-stabilizes the dynamics, ensuring that the iterates return to stability in a fixed number of iterations and allowing convergence in the best-iterate sense even with large learning rates. Experiments validate our theoretical findings and illustrate the benefits of SGD in the large-stepsize regime.

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