Exponential Concentration in Stochastic Approximation

Law, Kody J. H.; Walton, Neil; Yang, Shangda

Abstract

We analyze the behavior of stochastic approximation algorithms where iterates, in expectation, progress towards an objective at each step. When progress is proportional to the step size of the algorithm, we prove exponential concentration bounds. These tail-bounds contrast asymptotic normality results, which are more frequently associated with stochastic approximation. The methods that we develop rely on a proof of geometric ergodicity. This extends the result of Markov chains due to Hajek (1982) 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 non-vanishing gradient.

Citation

Law, K. J. H., Walton, N., & Yang, S. (online). Exponential Concentration in Stochastic Approximation. Operations Research, https://doi.org/10.1287/opre.2023.0425

• Journal Article Type: Article
• Acceptance Date: Feb 10, 2025
• Online Publication Date: Apr 22, 2025
• Deposit Date: May 1, 2025
• Publicly Available Date: May 7, 2025
• Journal: Operations Research
• Print ISSN: 0030-364X
• Electronic ISSN: 1526-5463
• Publisher: Institute for Operations Research and Management Sciences
• Peer Reviewed: Peer Reviewed
• DOI: https://doi.org/10.1287/opre.2023.0425
• Keywords: stochastic approximation, projected stochastic gradient descent, concentration bounds
• Public URL: https://durham-repository.worktribe.com/output/3805743