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We study policy gradient for mean-field control in continuous time in areinforcement learning setting. By considering randomised policies with entropyregularisation, we derive a gradient expectation representation of the valuefunction, which is amenable to actor-critic type algorithms, where the valuefunctions and the policies are learnt alternately based on observation samplesof the state and model-free estimation of the population state distribution,either by offline or online learning. In the linear-quadratic mean-fieldframework, we obtain an exact parametrisation of the actor and critic functionsdefined on the Wasserstein space. Finally, we illustrate the results of ouralgorithms with some numerical experiments on concrete examples.

Actor-Critic learning for mean-field control in continuous time