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 continuously 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 nontrivial extensions of them, to derive sharp rates of convergence. In particular, we show that the value functions for the N-particle control problems converge toward 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.