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December 2021 Randomized Hamiltonian Monte Carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates
George Deligiannidis, Daniel Paulin, Alexandre Bouchard-Côté, Arnaud Doucet
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Ann. Appl. Probab. 31(6): 2612-2662 (December 2021). DOI: 10.1214/20-AAP1659

Abstract

The bouncy particle sampler is a Markov chain Monte Carlo method based on a nonreversible piecewise deterministic Markov process. In this scheme, a particle explores the state space of interest by evolving according to a linear dynamics which is altered by bouncing on the hyperplane perpendicular to the gradient of the negative log-target density at the arrival times of an inhomogeneous poisson process (PP) and by randomly perturbing its velocity at the arrival times of a homogeneous PP. Under regularity conditions, we show here that the process corresponding to the first component of the particle and its corresponding velocity converges weakly towards a randomized Hamiltonian Monte Carlo (RHMC) process as the dimension of the ambient space goes to infinity. RHMC is another piecewise deterministic nonreversible Markov process where a Hamiltonian dynamics is altered at the arrival times of a homogeneous PP by randomly perturbing the momentum component. We then establish dimension-free convergence rates for RHMC for strongly log-concave targets with bounded Hessians using coupling ideas and hypocoercivity techniques. We use our understanding of the mixing properties of the limiting RHMC process to choose the refreshment rate parameter of BPS. This results in significantly better performance in our simulation study than previously suggested guidelines.

Funding Statement

This material is based upon work supported in part by the U.S. Army Research Laboratory and the U. S. Army Research Office, and by the U.K. Ministry of Defence (MoD) and the U.K. Engineering and Physical Research Council (EPSRC) under grant number EP/R013616/1 and by the EPSRC EP/R034710/1.

Acknowledgements

The authors would like to thank Peter Holderrieth for a careful reading of the manuscript and his invaluable suggestions and Philippe Gagnon for his insightful comments on the manuscript. G.D. would like to thank Gabriel Stoltz for many useful discussions. The authors would also like to thank the anonymous referees for numerous suggestions that have greatly improved the content and the presentation of the paper. A part of this research was done while A. Doucet, G. Deligiannidis and D. Paulin were hosted by the Institute for Mathematical Sciences in Singapore.

Citation

Download Citation

George Deligiannidis. Daniel Paulin. Alexandre Bouchard-Côté. Arnaud Doucet. "Randomized Hamiltonian Monte Carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates." Ann. Appl. Probab. 31 (6) 2612 - 2662, December 2021. https://doi.org/10.1214/20-AAP1659

Information

Received: 1 March 2020; Revised: 1 November 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/20-AAP1659

Subjects:
Primary: 60F17 , 65C05
Secondary: 60J25

Keywords: Bouncy particle sampler , coupling , hypocoercivity , Randomized Hamiltonian Monte Carlo , weak convergence

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 6 • December 2021
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