Collective Proposal Distributions for Nonlinear MCMC samplers: Mean-Field Theory and Fast Implementation

TL;DR

This paper introduces a nonlinear extension of the Metropolis-Hastings algorithm using mean-field theory, providing a stable, fast, and GPU-implementable method that outperforms classical approaches, especially on multimodal distributions.

Contribution

It develops a mean-field based nonlinear MCMC algorithm with convergence guarantees and demonstrates its efficiency and stability through GPU implementation and experiments.

Findings

Converges under double limit of iterations and particles

Faster convergence than classical methods

Stable performance on multimodal distributions

Abstract

Over the last decades, various "non-linear" MCMC methods have arisen. While appealing for their convergence speed and efficiency, their practical implementation and theoretical study remain challenging. In this paper, we introduce a non-linear generalization of the Metropolis-Hastings algorithm to a proposal that depends not only on the current state, but also on its law. We propose to simulate this dynamics as the mean field limit of a system of interacting particles, that can in turn itself be understood as a generalisation of the Metropolis-Hastings algorithm to a population of particles. Under the double limit in number of iterations and number of particles we prove that this algorithm converges. Then, we propose an efficient GPU implementation and illustrate its performance on various examples. The method is particularly stable on multimodal examples and converges faster than the classical methods.