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.
A Fisher-Rao gradient flow for entropic mean-field min-max games
Heriot-Watt University（赫瑞瓦特大学）；Maxwell Institute for Mathematical Sciences（麦克斯韦数学研究所）；University of Edinburgh（爱丁堡大学）；The Alan Turing Institute（阿兰·图灵研究所）；Simtopia（Simtopia公司）
The Advective Fisher-Rao Geometry of Deterministic Measure Transport
Institut für Mathematik, Technische Universität Berlin（柏林工业大学数学研究所）；Max-Planck-Institut für Mathematik in den Naturwissenschaften（马克斯·普朗克数学研究所（自然科学领域））
Fisher-Rao Gradient Flows of Linear Programs and State-Action Natural Policy Gradients
RWTH Aachen University（亚琛工业大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）
Center for Advanced Intelligence Project, RIKEN AIP, Japan（日本RIKEN AIP高级智能项目中心）；School of Mathematics, University of Edinburgh, UK（爱丁堡大学数学学院）；School of Mathematics, University of Edinburgh, UK, The Alan Turing Institute, UK（爱丁堡大学数学学院、英国艾伦·图灵研究所）
