A localized consensus-based sampling algorithm

We propose a localized consensus-based method for sampling from non-Gaussian distributions. This method arises from an alternative derivation of consensus-based sampling (CBS). Starting from ensemble-preconditioned Langevin dynamics, we approximate the potential with a Moreau envelope, replace the gradient in the Langevin equation with a proximal operator, and finally approximate this operator by a weighted mean. Under Gaussian initial and target distributions, this procedure recovers the standard CBS dynamics. In addition, when we retain only the approximations valid beyond the Gaussian case, we retrieve a refined variant of polarized CBS. The resulting algorithm, which we call localized consensus-based sampling, 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 fully gradient-free and easily parallelizable.