  We consider reinforcement learning (RL) methods for finding optimal policies
in linear quadratic (LQ) mean field control (MFC) problems over an infinite
horizon in continuous time, with common noise and entropy regularization. We
study policy gradient (PG) learning and first demonstrate convergence in a
model-based setting by establishing a suitable gradient domination
condition.Next, our main contribution is a comprehensive error analysis, where
we prove the global linear convergence and sample complexity of the PG
algorithm with two-point gradient estimates in a model-free setting with
unknown parameters. In this setting, the parameterized optimal policies are
learned from samples of the states and population distribution.Finally, we
provide numerical evidence supporting the convergence of our implemented
algorithms.
