Gaussian Approximation for Asynchronous Q-learning
This work provides theoretical guarantees for asynchronous Q-learning convergence, which is incremental as it builds on existing analysis with specific stepsize conditions.
The paper tackles the problem of analyzing the convergence rates for asynchronous Q-learning with polynomial stepsizes, establishing a rate of order up to n^{-1/6} log^4(nSA) for high-dimensional central limit theorem approximations over hyper-rectangles, where n is the number of samples and S and A are state and action counts. It also proves a related martingale difference theorem and provides bounds for high-order moments of the algorithm's last iterate.
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 $k^{-Ï},\, Ï\in (1/2, 1]$. 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.