  We propose a PDE-based accelerated gradient algorithm for optimal feedback
controls of McKean-Vlasov dynamics that involve mean-field interactions both in
the state and action. The method exploits a forward-backward splitting approach
and iteratively refines the approximate controls based on the gradients of
smooth costs, the proximal maps of nonsmooth costs, and dynamically updated
momentum parameters. At each step, the state dynamics is approximated via a
particle system, and the required gradient is evaluated through a coupled
system of nonlocal linear PDEs. The latter is solved by finite difference
approximation or neural network-based residual approximation, depending on the
state dimension. We present exhaustive numerical experiments for low and
high-dimensional mean-field control problems, including sparse stabilization of
stochastic Cucker-Smale models, which reveal that our algorithm captures
important structures of the optimal feedback control and achieves a robust
performance with respect to parameter perturbation.
