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arXiv · 2506.23335

Breaking a Logarithmic Barrier in the Stopping Time Convergence Rate of Stochastic First-order Methods

Yasong Feng·Yifan Jiang·Tianyu Wang·Zhiliang Ying
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Abstract

This work provides a novel convergence analysis for stochastic optimization in terms of stopping times, addressing the practical reality that algorithms are often terminated adaptively based on observed progress. Unlike prior approaches, our analysis: 1. Directly characterizes convergence in terms of stopping times adapted to the underlying stochastic process. 2. Breaks a logarithmic barrier in existing results. Key to our results is the development of a lemma to control the large deviation property of almost super-martingales. This lemma might be of broader interest.

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Yasong Feng, Yifan Jiang, Tianyu Wang, Zhiliang Ying. 2025-06-29. Breaking a Logarithmic Barrier in the Stopping Time Convergence Rate of Stochastic First-order Methods. https://arxiv.org/abs/2506.23335

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