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We study the convergence of $Q$-learning with linear function approximation.Our key contribution is the introduction of a novel multi-Bellman operator thatextends the traditional Bellman operator. By exploring the properties of thisoperator, we identify conditions under which the projected multi-Bellmanoperator becomes contractive, providing improved fixed-point guaranteescompared to the Bellman operator. To leverage these insights, we propose themulti $Q$-learning algorithm with linear function approximation. We demonstratethat this algorithm converges to the fixed-point of the projected multi-Bellmanoperator, yielding solutions of arbitrary accuracy. Finally, we validate ourapproach by applying it to well-known environments, showcasing theeffectiveness and applicability of our findings.

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