标题： 在随机一阶方法的停止时间收敛率中打破对数障碍 显示英文标题

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

摘要： 这项工作提供了关于停止时间的随机优化的新收敛分析，解决了算法通常根据观察到的进展自适应终止的实际现实。 与之前的方法不同，我们的分析：1. 直接根据适应于底层随机过程的停止时间来描述收敛性。 2. 打破了现有结果中的对数障碍。 我们结果的关键是开发了一种针对此类随机过程的格罗纳尔德类型论证。 这个工具能够在没有限制性假设的情况下提供更紧的界限。 显示英文摘要

摘要： 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 Gr\"onwall-type argument tailored to such stochastic processes. This tool enables sharper bounds without restrictive assumptions.