A localized consensus-based sampling algorithm

We propose a localized consensus-based method for sampling from non-Gaussian distributions, a task that frequently arises when solving Bayesian inverse problems. Our method arises from an alternative derivation of consensus-based sampling (CBS). Starting from ensemble-preconditioned Langevin dynamics, we replace the potential by its Moreau envelope -- a smoother approximation -- in order to replace the gradient in the Langevin equation with a proximal operator. We then approximate this operator by a weighted mean. In the limit of infinitely smoothing the potential to a quadratic function, this procedure recovers the standard CBS dynamics. In addition, outside this limit, we retrieve a refined variant of polarized CBS. We call the resulting algorithm localized consensus-based sampling, since particles interact more with nearby particles than with faraway ones. Our method is affine-invariant, exact for Gaussian targets in the mean-field limit, and demonstrates improved robustness over polarized CBS in numerical experiments. Like other consensus-based methods, localized CBS is gradient-free and easily parallelizable.