  Reinforcement learning is a powerful tool to learn the optimal policy of
possibly multiple agents by interacting with the environment. As the number of
agents grow to be very large, the system can be approximated by a mean-field
problem. Therefore, it has motivated new research directions for mean-field
control (MFC) and mean-field game (MFG). In this paper, we study the policy
gradient method for the linear-quadratic mean-field control and game, where we
assume each agent has identical linear state transitions and quadratic cost
functions. While most of the recent works on policy gradient for MFC and MFG
are based on discrete-time models, we focus on the continuous-time models where
some analyzing techniques can be interesting to the readers. For both MFC and
MFG, we provide policy gradient update and show that it converges to the
optimal solution at a linear rate, which is verified by a synthetic simulation.
For MFG, we also provide sufficient conditions for the existence and uniqueness
of the Nash equilibrium.
