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

TL;DR

This paper introduces a new convergence analysis for stochastic first-order methods that accounts for adaptive stopping times, overcoming previous logarithmic barriers and providing broader insights into algorithm termination behavior.

Contribution

It develops a novel analysis framework that directly characterizes convergence with adaptive stopping times, breaking the logarithmic barrier in existing results.

Findings

Breaks the logarithmic barrier in convergence rate analysis.

Introduces a lemma controlling large deviations of almost super-martingales.

Provides a more realistic analysis of stochastic optimization algorithms.

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.