# Convergence Rates of Time Discretization in Extended Mean Field Control

arXiv: https://arxiv.org/abs/2509.00904 · HTML: https://mlsanity.com/paper/2509.00904/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Aug 2025.

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

- [Actor-Critic Learning for Extended Mean Field Control with Deterministic Policies](https://mlsanity.com/paper/2607.11005/) (2026)

## Abstract (arXiv)

Piecewise constant control approximation provides a practical framework for designing numerical schemes of continuous-time control problems. We analyze the accuracy of such approximations for extended mean field control (MFC) problems, where the dynamics and costs depend on the joint distribution of states and controls. For linear-convex extended MFC problems, we show that the optimal control is 1/2-H"older continuous in time. Using this regularity, we prove that the optimal cost of the continuous-time problem can be approximated by piecewise constant controls with order 1/2, while the optimal control itself can be approximated with order 1/4. For general extended MFC problems, we further show that, under sufficient regularity of the value functions, the value functions converge with an improved first-order rate, matching the best-known rate for classical control problems without mean field interaction, and consistent with the numerical observations for MFC of Cucker-Smale models.