Mean-field Underdamped Langevin Dynamics and its Space-Time Discretization

A popular approach for finding a minimizer of an EMO problem is based on the mean-field Langevin dynamics. When the problem is convex, Hu et al. (2019) show the mean-field Langevin dynamics convergences to the minimizer asymptotically; and when the problem satisfies a uniform logarithmic Sobolev inequality, several works have established an exponentially fast convergence (Chizat, 2022; Nitanda et al., 2022; Chen et al., 2022). The mean-field Langevin dynamics, however, cannot be implemented and bridging this gap requires both space and time discretizations of the dynamics. A time-discretization of the mean-field Langevin dynamics was analyzed by Nitanda et al. (2022) who extend the interpolation argument introduced by Vempala and Wibisono (2019) to a non-linear Fokker-Planck equation. They establish the non-asymptotic convergence of the time-discretized dynamics in the energy gap. A space-discretization was studied by Chen et al. (2022) who show that the finite-particle approximation to the density of the mean-field Langevin dynamics (referred to as a uniform-in-time propagation of chaos) converges exponentially fast, with a bias related to the number of particles. More practically, Suzuki et al. (2023) analyze a space-time discretization of the mean-field Langevin dynamics and establish the non-asymptotic convergence 1