The optimal rate of convergence in mean field control via recoupled shadow flows

Abstract

We prove that the value functions of the $N$-particle stochastic optimal control problem converge uniformly to the value function of the corresponding mean field control problem at the empirical-measure rate ($N^{-1/d}$ for $d\geq3$ and $N^{-1/2}\sqrt{\log N}$ for $d=2$), for mean field costs that are merely Lipschitz continuous in the $1$-Wasserstein distance. For $d\geq2$ the rate is optimal in this class, which covers problems whose mean field optimizers are neither unique nor stable. This proves the rate conjectured by Daudin, Delarue and Jackson, and removes the semiconcavity hypothesis made there. The proof is control-theoretic: from each realization of an $N$-particle control we build a pathwise Fokker--Planck flow -- a shadow flow -- which, repeatedly recoupled to the particles by optimal transport, shadows the empirical measure at the empirical-measure rate. The same rate holds with additive common noise, uniformly in its intensity. Finally, in dimension one we discover that the empirical-measure benchmark is not optimal: cooperating particles beat it, and the optimal polynomial exponent is $4/7$, strictly between the accuracy of independent samples and that of quantization by freely placed points. The proofs combine an anticipating correction of the shadow flow, a Gibbs law implementing the cooperation, and a Schrödinger ground-state estimate for the optimality.