Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo

Christophe Andrieu; Alain Durmus; Nikolas Nüsken; Julien Roussel

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arXiv · 1808.08592

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 $\mathrm{L}^2$-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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Christophe Andrieu, Alain Durmus, Nikolas Nüsken, Julien Roussel. 2018-08-26. Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo. https://arxiv.org/abs/1808.08592

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