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

• URL: http://arxiv.org/abs/2309.16819v1
• Date: Thu, 28 Sep 2023 19:56:31 GMT
• Title: Multi-Bellman operator for convergence of $Q$-learning with linear function approximation
• Authors: Diogo S. Carvalho, Pedro A. Santos, Francisco S. Melo
• Abstract summary: We study the convergence of $Q$-learning with linear function approximation. By exploring the properties of a novel multi-Bellman operator, we identify conditions under which the projected multi-Bellman operator becomes contractive. We demonstrate that this algorithm converges to the fixed-point of the projected multi-Bellman operator, yielding solutions of arbitrary accuracy.
• Score: 3.6218162133579694
• License: http://creativecommons.org/licenses/by/4.0/
• 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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