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arXiv · 2602.16274

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

Rahul Singh·Siddharth Chandak·Eric Moulines·Vivek S. Borkar·Nicholas Bambos
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Abstract

We present the first regret bound for classical online Q-learning in infinite-horizon discounted Markov decision processes (MDPs), 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-$\tilde{O}(N^{9/10})$. We also obtain sample complexity guarantees, with both regret and sample complexity bounds holding with high probability. 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 framework is allowed to converge to one asymptotically.

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Rahul Singh, Siddharth Chandak, Eric Moulines, Vivek S. Borkar, Nicholas Bambos. 2026-02-18. Regret and Sample Complexity of Online Q-Learning via Concentration of Stochastic Approximation with Time-Inhomogeneous Markov Chains. https://arxiv.org/abs/2602.16274

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