Exponential Concentration in Stochastic Approximation

• 成果类型: Article; Early Access
• 署名作者: Law, Kody . T. H.; Walton, Neil; Yang, Shangda
• 署名单位: University of Manchester; Durham University
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2023.0425
• 发表日期: 2025
• 关键词: convergence

摘要

We analyze the behavior of stochastic approximation algorithms where iterates, in expectation, progress toward an objective at each step. When progress is proportional to the step size of the algorithm, we prove exponential concentration bounds. These tailbounds contrast asymptotic normality results, which are more frequently associated with stochastic approximation. The methods that we develop rely on a proof of geometric ergodicity. The extends results on the exponential ergodicity of Markov chains to stochastic approximation algorithms. We apply our results to several different stochastic approximation algorithms, specifically Projected Stochastic Gradient Descent, Kiefer-Wolfowitz, and Stochastic Frank-Wolfe algorithms. When applicable, our results prove faster O(1/t) and linear convergence rates for Projected Stochastic Gradient Descent with a nonvanishing gradient.