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

arXiv: https://arxiv.org/abs/2312.11373 · HTML: https://mlsanity.com/paper/2312.11373/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2023.

## Built on by (most cited first, 1 of 1)

- [Convergence analysis of controlled particle systems arising in deep learning: from finite to infinite sample size](https://mlsanity.com/paper/2404.05185/) (2024)

## Abstract (arXiv)

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.