Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo

Christophe Andrieu, Alain Durmus, Nikolas Nüsken, Julien Roussel

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Abstract

In this work, we establish L2-exponential convergence for a broad class of Piecewise Deterministic Markov Processes recently proposed in the context of Markov Process Monte Carlo methods and covering in particular the Randomized Hamiltonian Monte Carlo, the Zig-Zag process and the Bouncy Particle Sampler. The kernel of the symmetric part of the generator of such processes is non-trivial, and we follow the ideas recently introduced by (Dolbeault et al., 2009, 2015) to develop a rigorous framework for hypocoercivity in a fairly general and unifying set-up, while deriving tractable estimates of the constants involved in terms of the parameters of the dynamics. As a by-product we characterize the scaling properties of these algorithms with respect to the dimension of classes of problems, therefore providing some theoretical evidence to support their practical relevance.
Original languageEnglish
Pages (from-to)2478-2517
Number of pages40
JournalAnnals of Applied Probability
Volume31
Issue number5
DOIs
Publication statusPublished - 1 Oct 2021

Bibliographical note

Funding Information:
Acknowledgments. CA acknowledges support from EPSRC “Intractable Likelihood: New Challenges from Modern Applications (ILike)” (EP/K014463/1) and “Computational Statistical Inference for Engineering and Security (CoSInES)”, (EP/R034710/1). AD acknowledges support of the Lagrange Mathematical and Computing Research Center. JR would like to thank Pierre Monmarché for showing him how ZZ and BPS fall under a general framework. All the authors acknowledge the support of the Institute for Statistical Science in Bristol.

Publisher Copyright:
© 2021 Institute of Mathematical Statistics. All rights reserved.

Keywords

  • PDMCMC
  • geometric convergence
  • hypoellipticity

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