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arXiv · 2412.20553

Edge of Stochastic Stability: Revisiting the Edge of Stability for SGD

Arseniy Andreyev·Pierfrancesco Beneventano
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Abstract

Recent findings by Cohen et al., 2021, demonstrate that when training neural networks using full-batch gradient descent with a step size of $η$, the largest eigenvalue $λ_{\max}$ of the full-batch Hessian consistently stabilizes around $2/η$. These results have significant implications for convergence and generalization. This, however, is not the case for mini-batch optimization algorithms, limiting the broader applicabilityof the consequences of these findings. We show mini-batch Stochastic Gradient Descent (SGD) trains in a different regime we term Edge of Stochastic Stability (EoSS). In this regime, what stabilizes at $2/η$ is Batch Sharpness: the expected directional curvature of mini-batch Hessians along their corresponding stochastic gradients. As a consequence $λ_{\max}$ -- which is generally smaller than Batch Sharpness -- is suppressed, aligning with the long-standing empirical observation that smaller batches and larger step sizes favor flatter minima. We further discuss implications for mathematical modeling of SGD trajectories.

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Arseniy Andreyev, Pierfrancesco Beneventano. 2025-12-29. Edge of Stochastic Stability: Revisiting the Edge of Stability for SGD. https://arxiv.org/abs/2412.20553

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