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Multi-Bellman operator for convergence of $Q$-learning with linear function approximation (2309.16819v1)

Published 28 Sep 2023 in cs.LG and cs.AI

Abstract: We study the convergence of $Q$-learning with linear function approximation. Our key contribution is the introduction of a novel multi-BeLLMan operator that extends the traditional BeLLMan operator. By exploring the properties of this operator, we identify conditions under which the projected multi-BeLLMan operator becomes contractive, providing improved fixed-point guarantees compared to the BeLLMan operator. To leverage these insights, we propose the multi $Q$-learning algorithm with linear function approximation. We demonstrate that this algorithm converges to the fixed-point of the projected multi-BeLLMan operator, yielding solutions of arbitrary accuracy. Finally, we validate our approach by applying it to well-known environments, showcasing the effectiveness and applicability of our findings.

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