Continuous-Time q-learning for McKean-Vlasov Control Problems

This paper studies the q-learning, recently coined as the continuous-time counterpart of Q-learning by Jia and Zhou (2022c), for continuous time Mckean-Vlasov control problems in the setting of entropy-regularized reinforcement learning. In contrast to the single agent's control problem in Jia and Zhou (2022c), the mean-field interaction of agents render the definition of q-function more subtle, for which we reveal that two distinct q-functions naturally arise: (i) the integrated q-function (denoted by $q$) as the first-order approximation of the integrated Q-function introduced in Gu, Guo, Wei and Xu (2023) that can be learnt by a weak martingale condition involving test policies; and (ii) the essential q-function (denoted by $q_e$) that is employed in the policy improvement iterations. We show that two q-functions are related via an integral representation under all test policies. Based on the weak martingale condition of the integrated q-function and our proposed searching method of test policies, some model-free offline and online learning algorithms are devised. In two financial applications, one in LQ control framework and one beyond LQ control framework, we can obtain the exact parameterization of the value function and two q-functions and illustrate our algorithms with simulation experiments.