# Entropic Fictitious Play for Mean Field Optimization Problem

arXiv: https://arxiv.org/abs/2202.05841 · HTML: https://mlsanity.com/paper/2202.05841/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2022.

## Built on by (most cited first, 12 of 12)

- [Uniform-in-time propagation of chaos for mean field Langevin dynamics](https://mlsanity.com/paper/2212.03050/) (2022)
- [Improved Particle Approximation Error for Mean Field Neural Networks](https://mlsanity.com/paper/2405.15767/) (2024)
- [Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems](https://mlsanity.com/paper/2312.01127/) (2023)
- [Global Convergence of SGD On Two Layer Neural Nets](https://mlsanity.com/paper/2210.11452/) (2022)
- [Direct Distributional Optimization for Provable Alignment of Diffusion Models](https://mlsanity.com/paper/2502.02954/) (2025)
- [Propagation of Chaos for Mean-Field Langevin Dynamics and its Application to Model Ensemble](https://mlsanity.com/paper/2502.05784/) (2025)
- [Primal and Dual Analysis of Entropic Fictitious Play for Finite-sum Problems](https://mlsanity.com/paper/2303.02957/) (2023)
- [Linear convergence of proximal descent schemes on the Wasserstein space](https://mlsanity.com/paper/2411.15067/) (2024)
- [Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets](https://mlsanity.com/paper/2309.09258/) (2023)
- [Non-convex entropic mean-field optimization via Best Response flow](https://mlsanity.com/paper/2505.22760/) (2025)
- [Stochastic Control Methods for Optimization](https://mlsanity.com/paper/2601.01248/) (2026)
- [On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization](https://mlsanity.com/paper/2602.15094/) (2026)

## Abstract (arXiv)

We study two-layer neural networks in the mean field limit, where the number of neurons tends to infinity. In this regime, the optimization over the neuron parameters becomes the optimization over the probability measures, and by adding an entropic regularizer, the minimizer of the problem is identified as a fixed point. We propose a novel training algorithm named entropic fictitious play, inspired by the classical fictitious play in game theory for learning Nash equilibriums, to recover this fixed point, and the algorithm exhibits a two-loop iteration structure. Exponential convergence is proved in this paper and we also verify our theoretical results by simple numerical examples.