  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.
