A General-Purpose Theorem for High-Probability Bounds of Stochastic Approximation with Polyak Averaging
This work addresses a theoretical gap for researchers in optimization and reinforcement learning, providing a foundational tool for analyzing averaged stochastic algorithms, though it is incremental as it builds on existing averaging techniques.
The paper tackles the lack of high-probability performance guarantees for Polyak-Ruppert averaging in stochastic approximation algorithms by presenting a general framework for deriving non-asymptotic concentration bounds on averaged iterates, showing tightness up to constant factors and applying it to algorithms like temporal difference learning and Q-learning.
Polyak-Ruppert averaging is a widely used technique to achieve the optimal asymptotic variance of stochastic approximation (SA) algorithms, yet its high-probability performance guarantees remain underexplored in general settings. In this paper, we present a general framework for establishing non-asymptotic concentration bounds for the error of averaged SA iterates. Our approach assumes access to individual concentration bounds for the unaveraged iterates and yields a sharp bound on the averaged iterates. We also construct an example, showing the tightness of our result up to constant multiplicative factors. As direct applications, we derive tight concentration bounds for contractive SA algorithms and for algorithms such as temporal difference learning and Q-learning with averaging, obtaining new bounds in settings where traditional analysis is challenging.