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.

翻译：在过去几十年中,出现了各种“非线性”MCMC方法。在呼吁它们的趋同速度和效率的同时,它们的实际执行和理论研究仍然具有挑战性。在本文中,我们引入了大都会-哈斯廷斯算法的非线性概括,以作为不仅取决于当前状态,而且取决于其法律的提案。我们提议将这种动态模拟为互动粒子系统的平均场限制,这反过来又可以被理解为大都会-哈斯廷斯算法对粒子群的概括。在迭代数和粒子数的双重限制下,我们证明这种算法是趋同的。然后,我们提出高效的 GPU 实施,并用各种例子来说明其表现。这种方法在多式例子上特别稳定,比经典方法更快。