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

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\leq1. 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(ε^{-1}) synchronous iterations suffice for an ε-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.