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 $\mathcal{U} :[0,T] \times \mathcal{P}_2(\mathbb{R}^d) \to \mathbb{R}$ is Lipschitz, but may not be differentiable. In this setting, the first and last named authors recently identified an open and dense subset $\mathcal{O}$ of $[0,T] \times \mathcal{P}_2(\mathbb{R}^d)$ on which $\mathcal{U}$ is $\mathcal{C}^1$ 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 $\mathcal{U}$ with a rate of $1/N$, uniformly on subsets of $\mathcal{O}$ 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 $\mathcal{U}$ 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 $\mathcal{O}$ 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.