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.
Funding Statement
This research was conducted while A.D. was supported by an EPSRC-Roth scholarship co-funded by the Engineering and Physical Sciences Research Council and the Department of Mathematics at Imperial College London.
Acknowledgments
The authors wish to thank Pierre Degond, Robin Ryder and Christian Robert for their support and useful advice. A.D. acknowledges the hospitality of the CEREMADE, Université Paris-Dauphine where part of this research was carried out. G.C. acknowledges the hospitality of the Mathematics Department at Imperial College London where part of this research was carried out.
Citation
Grégoire Clarté. Antoine Diez. Jean Feydy. "Collective proposal distributions for nonlinear MCMC samplers: Mean-field theory and fast implementation." Electron. J. Statist. 16 (2) 6395 - 6460, 2022. https://doi.org/10.1214/22-EJS2091
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