On the mean field limit of consensus based methods

TL;DR

This paper analyzes the behavior of consensus-based optimization and sampling methods as the number of particles grows large, establishing their mean field limits and the well-posedness of the associated equations.

Contribution

It rigorously derives the mean field limit for consensus-based methods, proving existence and uniqueness of solutions for the limiting equations.

Findings

Existence of a unique strong solution for finite-particle SDEs.

Derivation of a Fokker-Planck equation in the mean-field limit.

Uniqueness of the solution to the McKean-Vlasov SDE.

Abstract

Consensus based optimization (CBO) employs a swarm of particles evolving as a system of stochastic differential equations (SDEs). Recently, it has been adapted to yield a derivative free sampling method referred to as consensus based sampling (CBS). In this paper, we investigate the ``mean field limit'' of a class of consensus methods, including CBO and CBS. This limit allows to characterize the system's behavior as the number of particles approaches infinity. Building upon prior work such as (Huang and Qiu, 2022), we establish the existence of a unique, strong solution for these finite-particle SDEs. We further provide uniform moment estimates, which allow to show a Fokker-Planck equation in the mean-field limit. Finally, we prove that the limiting McKean-Vlasov type SDE related to the Fokker-Planck equation admits a unique solution.