Regret and Sample Complexity of Online Q-Learning via Concentration of Stochastic Approximation with Time-Inhomogeneous Markov Chains

arXiv:2602.16274v1 Announce Type: cross
Abstract: We present the first high-probability regret bound for classical online Q-learning in infinite-horizon discounted Markov decision processes, without relying on optimism or bonus terms. We first analyze Boltzmann Q-learning with decaying temperature and show that its regret depends critically on the suboptimality gap of the MDP: for sufficiently large gaps, the regret is sublinear, while for small gaps it deteriorates and can approach linear growth. To address this limitation, we study a Smoothed $epsilon_n$-Greedy exploration scheme that combines $epsilon_n$-greedy and Boltzmann exploration, for which we prove a gap-robust regret bound of near-$tilde{O}(N^{9/10})$. To analyze these algorithms, we develop a high-probability concentration bound for contractive Markovian stochastic approximation with iterate- and time-dependent transition dynamics. This bound may be of independent interest as the contraction factor in our bound is governed by the mixing time and is allowed to converge to one asymptotically.