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arXiv · 2610.20347

Non-asymptotic Convergence of Average-reward Q-learning with Options

Kintan Saha·Ahana Deb·Anders Jonsson·Debabrota Basu
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Abstract

We study the finite-sample convergence of Q-learning algorithms for the average-reward formulation of Reinforcement Learning (RL) with options. While options allow temporarily extended and hierarchical actions, the average-reward formulation incorporates continual tasks beyond discounted and episodic RL. We establish the first $\tilde{O}(1/\text{iterations})$ convergence guarantees, in the mean-square sense, for average-reward option Q-learning algorithms in both the inter- and intra-option settings. First, we prove that both inter- and intra-option Bellman operators satisfy multi-step span contractions under unichain assumptions. Then, we propose synchronous estimators of these multi-step option Bellman operators leading to two variants of option Q-learning. Finally, we extend the stochastic approximation framework to prove convergence rates of these two algorithms under the minimal assumptions used in the literature.

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Kintan Saha, Ahana Deb, Anders Jonsson, Debabrota Basu. 2026-10-14. Non-asymptotic Convergence of Average-reward Q-learning with Options. https://arxiv.org/abs/2610.20347

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