Finite-Time Analysis of Asynchronous Stochastic Approximation and $Q$-Learning

TL;DR

This paper provides a finite-time convergence analysis for asynchronous stochastic approximation schemes, including $Q$-learning, achieving bounds that match or improve upon existing results for both synchronous and asynchronous cases.

Contribution

It introduces a general asynchronous SA framework with a weighted infinity-norm contractive operator and derives sharp finite-time bounds, specifically improving asynchronous $Q$-learning analysis.

Findings

Finite-time convergence bounds for asynchronous SA schemes.

Matching the best bounds for synchronous $Q$-learning.

Improved bounds for asynchronous $Q$-learning.

Abstract

We consider a general asynchronous Stochastic Approximation (SA) scheme featuring a weighted infinity-norm contractive operator, and prove a bound on its finite-time convergence rate on a single trajectory. Additionally, we specialize the result to asynchronous $Q$-learning. The resulting bound matches the sharpest available bound for synchronous $Q$-learning, and improves over previous known bounds for asynchronous $Q$-learning.