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^-ω, ω ∈ (1/2, 1]. Assuming that the sequence of state-action-next-state triples (s_k, a_k, s_k+1)_k ≥ 0 forms a uniformly geometrically ergodic Markov chain, we establish a rate of order up to n^-1/6 log⁴ (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.