Uniform-in-$N$ log-Sobolev inequality for the mean-field Langevin dynamics with convex energy

TL;DR

This paper proves a uniform log-Sobolev inequality for mean-field Langevin dynamics, showing the constant does not depend on the number of particles, using a Lipschitz transport map and reverse heat flow techniques.

Contribution

It introduces a novel uniform-in-$N$ log-Sobolev inequality for mean-field Langevin dynamics with convex energy functions.

Findings

Established a uniform log-Sobolev inequality independent of particle number $N$

Developed a Lipschitz transport map via reverse heat flow

Provided new tools for analyzing mean-field stochastic processes

Abstract

We establish a log-Sobolev inequality for the stationary distribution of mean-field Langevin dynamics with a constant that is independent of the number of particles $N$. Our proof proceeds by establishing the existence of a Lipschitz transport map from the standard Gaussian measure via the reverse heat flow of Kim and Milman.