  We develop a new policy gradient and actor-critic algorithm for solving
mean-field control problems within a continuous time reinforcement learning
setting. Our approach leverages a gradient-based representation of the value
function, employing parametrized randomized policies. The learning for both the
actor (policy) and critic (value function) is facilitated by a class of moment
neural network functions on the Wasserstein space of probability measures, and
the key feature is to sample directly trajectories of distributions. A central
challenge addressed in this study pertains to the computational treatment of an
operator specific to the mean-field framework. To illustrate the effectiveness
of our methods, we provide a comprehensive set of numerical results. These
encompass diverse examples, including multi-dimensional settings and nonlinear
quadratic mean-field control problems with controlled volatility.
