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arXiv · 2309.16819

Multi-Bellman operator for convergence of $Q$-learning with linear function approximation

Diogo S. Carvalho·Pedro A. Santos·Francisco S. Melo
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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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Diogo S. Carvalho, Pedro A. Santos, Francisco S. Melo. 2023-09-28. Multi-Bellman operator for convergence of $Q$-learning with linear function approximation. https://arxiv.org/abs/2309.16819

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