Piecewise deterministic sampling with splitting schemes

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.

References in corpus (20)

• Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps
• Rejection-free Monte-Carlo sampling for general potentials
• Irreversible Monte Carlo Algorithms for Efficient Sampling
• The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics
• Underdamped Langevin MCMC: A non-asymptotic analysis
• Stochastic resonance-free multiple time-step algorithm for molecular dynamics with very large time steps
• Piecewise-Deterministic Markov Chain Monte Carlo
• Ergodicity of the zigzag process
• Piecewise Deterministic Markov Processes for Scalable Monte Carlo on Restricted Domains
• Probabilistic and Piecewise Deterministic models in Biology
• Pointwise Gradient Bounds for Degenerate Semigroups (of UFG type)
• High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm
• Sticky PDMP samplers for sparse and local inference problems
• Second order quantitative bounds for unadjusted generalized Hamiltonian Monte Carlo
• Mixing of Metropolis-Adjusted Markov Chains via Couplings: The High Acceptance Regime
• Randomized Hamiltonian Monte Carlo as Scaling Limit of the Bouncy Particle Sampler and Dimension-Free Convergence Rates
• NuZZ: numerical Zig-Zag sampling for general models
• Nonreversible MCMC from conditional invertible transforms: a complete recipe with convergence guarantees
• PDMP Monte Carlo methods for piecewise-smooth densities
• Weak error expansion of a stopped numerical scheme for singular Langevin process