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

2026-02-16

AI Summary

This work investigates optimization problems over spaces of probability measures, inspired by mean-field theories of neural networks. To this end, we construct a kernel-smoothed Fisher–Rao gradient flow and introduce a corresponding interacting particle system in its mean-field limit. We establish, for the first time, existence and uniqueness of solutions for this kernelized Fisher–Rao flow and prove the propagation of chaos for the associated particle system. These results not only guarantee the well-posedness and convergence of the particle approximation but also yield a theoretically grounded and computationally tractable algorithm for entropy-based mean-field optimization.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• propagation of chaos
• Fisher-Rao gradient flow
• mean-field optimization
• interacting particle system
• entropic optimization

Innovation

• Fisher-Rao gradient flow
• propagation of chaos
• mean-field optimization
• interacting particle system
• kernelized energy functional