Sharp convergence rates for mean field control in the region of strong regularity

We study the convergence problem for mean field control, also known as optimal control of McKean-Vlasov dynamics. We assume that the data is smooth but not convex, and thus the limiting value function U :[0,T] × P2(ℝd) → ℝ is Lipschitz, but may not be differentiable. In this setting, the first and last named authors recently identified an open and dense subset O of [0,T] × P2(ℝd) on which U is C1 and solves the relevant infinite-dimensional Hamilton-Jacobi equation in a classical sense. In the present paper, we use these regularity results, and some non-trivial extensions of them, to derive sharp rates of convergence. In particular, we show that the value functions for the N-particle control problems converge towards U with a rate of 1/N, uniformly on subsets of O which are compact in the p-Wasserstein space for some p > 2. A similar result is also established at the level of the optimal feedback controls. The rate 1/N is the optimal rate in this setting even if U is smooth, while, in general, the optimal global rate of convergence is known to be slower than 1/N. Thus our results show that the rate of convergence is faster inside of O than it is outside. As a consequence of the convergence of the optimal feedbacks, we obtain a concentration inequality for optimal trajectories of the N-particle problem started from i.i.d. initial conditions.