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Over the last decades, various "non-linear" MCMC methods have arisen. Whileappealing for their convergence speed and efficiency, their practicalimplementation and theoretical study remain challenging. In this paper, weintroduce a non-linear generalization of the Metropolis-Hastings algorithm to aproposal that depends not only on the current state, but also on its law. Wepropose to simulate this dynamics as the mean field limit of a system ofinteracting particles, that can in turn itself be understood as ageneralisation of the Metropolis-Hastings algorithm to a population ofparticles. Under the double limit in number of iterations and number ofparticles we prove that this algorithm converges. Then, we propose an efficientGPU implementation and illustrate its performance on various examples. Themethod is particularly stable on multimodal examples and converges faster thanthe classical methods.

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