Sharp propagation of chaos for the ensemble Langevin sampler

Abstract

The aim of this paper is to revisit propagation of chaos for a Langevin-type interacting particle system recently proposed as a method to sample probability measures. The interacting particle system we consider coincides, in the setting of a log-quadratic target distribution, with the ensemble Kalman sampler [SIAM J. Appl. Dyn. Syst. 19 (2020), no. 1, 412–441], for which propagation of chaos was first proved by Ding and Li in [SIAM J. Math. Anal. 53 (2021), no. 2, 1546–1578]. Like these authors, we prove propagation of chaos with an approach based on a synchronous coupling, as in Sznitman's classical argument. Instead of relying on a boostrapping argument, however, we use a technique based on stopping times in order to handle the presence of the empirical covariance in the coefficients of the dynamics. The use of stopping times to handle the lack of global Lipschitz continuity in the coefficients of stochastic dynamics originates from numerical analysis [SIAM J. Numer. Anal. 40 (2002), no. 3, 1041–1063] and was recently employed to prove mean-field limits for consensus-based optimization and related interacting particle systems [arXiv:2312.07373, 2023; Math. Models Methods Appl. Sci. 33 (2023), no. 2, 289–339]. In the context of ensemble Langevin sampling, this technique enables proving pathwise propagation of chaos with optimal rate, whereas previous results were optimal only up to a positive $\epsilon$.