# On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization

arXiv: https://arxiv.org/abs/2602.15094 · HTML: https://mlsanity.com/paper/2602.15094/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 54 references that are in the map.
- First posted on arXiv 16 Feb 2026.

## Builds on (most cited first, 21 of 21)

- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019)
- [Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM](https://mlsanity.com/paper/2206.08873/) (2022)
- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [A mean-field analysis of two-player zero-sum games](https://mlsanity.com/paper/2002.06277/) (2020)
- [Uniform-in-time propagation of chaos for mean field Langevin dynamics](https://mlsanity.com/paper/2212.03050/) (2022)
- [A Fisher-Rao gradient flow for entropy-regularised Markov decision processes in Polish spaces](https://mlsanity.com/paper/2310.02951/) (2023)
- [Convergence of Policy Gradient for Entropy Regularized MDPs with Neural Network Approximation in the Mean-Field Regime](https://mlsanity.com/paper/2201.07296/) (2022)
- [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024)
- [Entropic Fictitious Play for Mean Field Optimization Problem](https://mlsanity.com/paper/2202.05841/) (2022)
- [Improved Particle Approximation Error for Mean Field Neural Networks](https://mlsanity.com/paper/2405.15767/) (2024)
- [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024)
- [Propagation of Chaos for Mean-Field Langevin Dynamics and its Application to Model Ensemble](https://mlsanity.com/paper/2502.05784/) (2025)
- [Linear convergence of proximal descent schemes on the Wasserstein space](https://mlsanity.com/paper/2411.15067/) (2024)
- [A Fisher-Rao gradient flow for entropic mean-field min-max games](https://mlsanity.com/paper/2405.15834/) (2024)
- [Mirror Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2505.02621/) (2025)
- [Non-convex entropic mean-field optimization via Best Response flow](https://mlsanity.com/paper/2505.22760/) (2025)

## Abstract (arXiv)

We consider a class of optimization problems on the space of probability measures motivated by the mean-field approach to studying neural networks. Such problems can be solved by constructing continuous-time gradient flows that converge to the minimizer of the energy function under consideration, and then implementing discrete-time algorithms that approximate the flow. In this work, we focus on the Fisher-Rao gradient flow and we construct an interacting particle system that approximates the flow as its mean-field limit. We discuss the connection between the energy function, the gradient flow and the particle system and explain different approaches to smoothing out the energy function with an appropriate kernel in a way that allows for the particle system to be well-defined. We provide a rigorous proof of the existence and uniqueness of thus obtained kernelized flows, as well as a propagation of chaos result that provides a theoretical justification for using the corresponding kernelized particle systems as approximation algorithms in entropic mean-field optimization.