Continuous Time q-Learning for Mean-Field Control Problems

This paper studies the q-learning, recently coined as the continuous time counterpart of Q-learning by Jia and Zhou (J Mach Learn Res 24:1–61, 2023), for continuous time mean-field control problems in the setting of entropy-regularized reinforcement learning. In contrast to the single agent’s control problem in Jia and Zhou (J Mach Learn Res 24:1–61, 2023), we reveal that two different q-functions naturally arise in mean-field control problems: (i) the integrated q-function (denoted by q) as the first-order approximation of the integrated Q-function introduced in Gu et al. (Oper Res 71(4):1040–1054, 2023), which can be learnt by a weak martingale condition using all test policies; and (ii) the essential q-function (denoted by \(q_e\) $q_e$ ) that is employed in the policy improvement iterations. We show that two q-functions are related via an integral representation. Based on the weak martingale condition and our proposed searching method of test policies, some model-free learning algorithms are devised. In two examples, one in LQ control framework and one beyond LQ control framework, we can obtain the exact parameterization of the optimal value function and q-functions and illustrate our algorithms with simulation experiments.