Exponential Concentration in Stochastic Approximation
Kody J. H. Law, Neil Walton, Shangda Yang
Operations Research · 2026 · Volume 74 · Issue 1
Abstract
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 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. 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 ) O ( 1 / t ) and linear convergence rates for Projected Stochastic Gradient Descent with a nonvanishing gradient. Funding: This work was supported by EPSRC INFORMED-AI (EP/Y028732/1). Supplemental Material: The E-companion is available at https://doi.org/10.1287/opre.2023.0425 .
Keywords
Optimization, concentration bounds, stochastic approximation, projected stochastic gradient descent
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