# Learning Mean-Field Games through Mean-Field Actor-Critic Flow

arXiv: https://arxiv.org/abs/2510.12180 · HTML: https://mlsanity.com/paper/2510.12180/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- First posted on arXiv 14 Oct 2025.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Built on by (most cited first, 2 of 2)

- [High-dimensional Mean-Field Games by Particle-based Flow Matching](https://mlsanity.com/paper/2512.01172/) (2025)
- [Optimal Transport Meets Reinforcement Learning: A Survey](https://mlsanity.com/paper/2610.01413/) (2026)

## Abstract (arXiv)

We propose the Mean-Field Actor-Critic (MFAC) flow, a continuous-time learning dynamics for solving mean-field games (MFGs), combining techniques from reinforcement learning and optimal transport. The MFAC framework jointly evolves the control (actor), value function (critic), and distribution components through coupled gradient-based updates governed by partial differential equations (PDEs). A central innovation is the Optimal Transport Geodesic Picard (OTGP) flow, which drives the distribution toward equilibrium along Wasserstein-2 geodesics. We conduct a rigorous convergence analysis using Lyapunov functionals and establish global exponential convergence of the MFAC flow under a suitable timescale. Our results highlight the algorithmic interplay among actor, critic, and distribution components. Numerical experiments illustrate the theoretical findings and demonstrate the effectiveness of the MFAC framework in computing MFG equilibria.