Localized consensus-based sampling approximates a proximal operator with a weighted mean, yielding an affine-invariant sampler that is exact for Gaussian targets and robust on multimodal targets in experiments.
On the mean field limit of consensus based methods
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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.
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A localized consensus-based sampling algorithm
Localized consensus-based sampling approximates a proximal operator with a weighted mean, yielding an affine-invariant sampler that is exact for Gaussian targets and robust on multimodal targets in experiments.