Convergence, Sticking and Escape: Stochastic Dynamics Near Critical Points in SGD
Abstract
We study the convergence properties and escape dynamics of Stochastic Gradient Descent (SGD) in one-dimensional landscapes, separately considering infinite- and finite-variance noise. Our main focus is to identify the time scales on which SGD reliably moves from an initial point to the local minimum in the same ''basin''. Under suitable conditions on the noise distribution, we prove that SGD converges to the basin's minimum unless the initial point lies too close to a local maximum. In that near-maximum scenario, we show that SGD can linger for a long time in its neighborhood. For initial points near a ''sharp'' maximum, we show that SGD does not remain stuck there, and we provide results to estimate the probability that it will reach each of the two neighboring minima. Overall, our findings present a nuanced view of SGD's transitions between local maxima and minima, influenced by both noise characteristics and the underlying function geometry.
Keywords
Cite
@article{arxiv.2505.18535,
title = {Convergence, Sticking and Escape: Stochastic Dynamics Near Critical Points in SGD},
author = {Dmitry Dudukalov and Artem Logachov and Vladimir Lotov and Timofei Prasolov and Evgeny Prokopenko and Anton Tarasenko},
journal= {arXiv preprint arXiv:2505.18535},
year = {2026}
}
Comments
The introduction, Subsections 2.1 ("Suitable Time Scaling") and 2.2 ("Sticking to a Critical Point"), as well as a small portion of the proof, have been revised. Subsection 2.3 ("Leaving the Neighborhood of a Sharp Maximum") has undergone minor revisions due to the equality in the doubly exponential case