Uniform-in-\({N}\) Log-Sobolev Inequality for the Mean-Field Langevin Dynamics with Convex Energy

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 [Formula: see text]. 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.