Piecewise deterministic sampling with splitting schemes

TL;DR

This paper introduces novel MCMC algorithms based on piecewise-deterministic processes and splitting schemes, achieving high accuracy with low computational cost, and demonstrates their effectiveness through theoretical analysis and numerical experiments.

Contribution

It develops unadjusted and adjusted splitting scheme-based MCMC algorithms for piecewise-deterministic processes, with second-order weak error and practical guidelines for optimal structure.

Findings

Unadjusted schemes have second-order weak error in step size.

Algorithms maintain low computational cost with one gradient evaluation per iteration.

Numerical experiments show promising results in Bayesian imaging and particle systems.

Abstract

We introduce Markov chain Monte Carlo (MCMC) algorithms based on numerical approximations of piecewise-deterministic Markov processes obtained with the framework of splitting schemes. We present unadjusted as well as adjusted algorithms, for which the asymptotic bias due to the discretisation error is removed applying a non-reversible Metropolis-Hastings filter. In a general framework we demonstrate that the unadjusted schemes have weak error of second order in the step size, while typically maintaining a computational cost of only one gradient evaluation of the negative log-target function per iteration. Focusing then on unadjusted schemes based on the Bouncy Particle and Zig-Zag samplers, we provide conditions ensuring geometric ergodicity and consider the expansion of the invariant measure in terms of the step size. We analyse the dependence of the leading term in this expansion on the refreshment rate and on the structure of the splitting scheme, giving a guideline on which structure is best. Finally, we illustrate promising results for our samplers with numerical experiments on a Bayesian imaging inverse problem and a system of interacting particles.