$O(1/n)$ Convergence Rate for RVI $Q$-Learning

Abstract

We establish finite-time convergence guarantee for synchronous relative value iteration (RVI) $Q$-learning for finite weakly communicating Markov decision processes (MDPs). For stochastic Krasnosel'ski\uı--Mann iterations driven by a non-expansive, linearly regular map on a finite-dimensional normed space, we establish last-iterate mean-squared residual bounds under martingale-difference noise whose conditional second moment grows at most linearly with the squared residual. Suitable polynomial stepsizes of order $n^{-a}$ yield an $O(n^{-a})$ rate for $0<a\le 1$. We verify linear regularity and the noise condition for the centered Bellman operator for finite weakly communicating MDPs. Applying the general result gives an $O(n^{-1})$ bound on the mean-squared Bellman span residual of RVI $Q$-learning iterates when a suitable stepsize sequence is used. The positive gap between optimal and suboptimal stationary deterministic policies then yields an $O(n^{-1})$ bound on the expected suboptimality of the greedy policy. Consequently, $O({\epsilon }^{-1})$ synchronous iterations suffice for an $\epsilon$-optimal policy in expectation. These guarantees are instance-dependent, with constants that depend on the hitting-time parameters and the suboptimality gap of the second-best policy.

arXiv comment24 pages