Gaussian Approximation for Asynchronous Q-learning

TL;DR

This paper establishes convergence rates and a high-dimensional CLT for asynchronous Q-learning with polynomial stepsizes, under geometric ergodicity assumptions, contributing new theoretical insights into its statistical properties.

Contribution

It derives convergence rates and a high-dimensional CLT for asynchronous Q-learning, including bounds for moments of the last iterate, under geometric ergodicity.

Findings

Convergence rate up to n^{-1/6} log^{4}(n S A) for the algorithm.

High-dimensional CLT for sums of martingale differences.

Bounds for high-order moments of the last iterate.

Abstract

In this paper, we derive rates of convergence in the high-dimensional central limit theorem for Polyak-Ruppert averaged iterates generated by the asynchronous Q-learning algorithm with a polynomial stepsize $k^{-\omega},\, \omega \in (1/2, 1]$. Assuming that the sequence of state-action-next-state triples $(s_k, a_k, s_{k+1})_{k \geq 0}$ forms a uniformly geometrically ergodic Markov chain, we establish a rate of order up to $n^{-1/6} \log^{4} (nS A)$ over the class of hyper-rectangles, where $n$ is the number of samples used by the algorithm and $S$ and $A$ denote the numbers of states and actions, respectively. To obtain this result, we prove a high-dimensional central limit theorem for sums of martingale differences, which may be of independent interest. Finally, we present bounds for high-order moments for the algorithm's last iterate.