Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo

Christophe Andrieu; Alain Durmus; Nikolas Nüsken; Julien Roussel

doi:10.1214/20-AAP1653

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 language	English
Pages (from-to)	2478-2517
Number of pages	40
Journal	Annals of Applied Probability
Volume	31
Issue number	5
DOIs	https://doi.org/10.1214/20-AAP1653
Publication status	Published - 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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Professor Christophe Andrieu
- C.Andrieubristol.acuk
- Statistical Science
- Probability, Analysis and Dynamics
- School of Mathematics - Professor in Statistics
- Statistics
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Cite this

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title = "Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo",

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. ",

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author = "Christophe Andrieu and Alain Durmus and Nikolas N{\"u}sken and Julien Roussel",

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 Monmarch{\'e} 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: {\textcopyright} 2021 Institute of Mathematical Statistics. All rights reserved.",

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AU - Durmus, Alain

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N1 - 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.

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N2 - 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.

AB - 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.

KW - PDMCMC

KW - geometric convergence

KW - hypoellipticity

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DO - 10.1214/20-AAP1653

M3 - Article (Academic Journal)

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JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 5

ER -

Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo

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