On explicit $L^2$-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms

Abstract

We establish $L^2$-exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process, and the bouncy particle sampler. Our analysis is based on a variational framework for hypocoercivity, which combines a Poincaré-type inequality in time-augmented state space and a standard $L^2$ energy estimate. Our analysis provides explicit convergence rate estimates, which are more quantitative than existing results.

arXiv commentUnder minor revision

References in corpus (19)

• The Zig-Zag Process and Super-Efficient Sampling for Bayesian Analysis of Big Data
• Rejection-free Monte-Carlo sampling for general potentials
• Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus
• Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation
• Randomized Hamiltonian Monte Carlo
• Geometric integrators and the Hamiltonian Monte Carlo method
• Non-reversible Metropolis-Hastings
• Piecewise-Deterministic Markov Chain Monte Carlo
• Piecewise Deterministic Markov Processes for Continuous-Time Monte Carlo
• Ergodicity of the zigzag process
• Piecewise Deterministic Markov Processes for Scalable Monte Carlo on Restricted Domains
• Hypocoercivity for Kolmogorov backward evolution equations and applications
• Randomized Hamiltonian Monte Carlo as Scaling Limit of the Bouncy Particle Sampler and Dimension-Free Convergence Rates
• Generalized Bouncy Particle Sampler
• The Coordinate Sampler: A Non-Reversible Gibbs-like MCMC Sampler
• Spectral analysis of the zigzag process
• Hypocoercivity with Schur complements
• High-dimensional scaling limits of piecewise deterministic sampling algorithms
• Posterior computation with the Gibbs zig-zag sampler

Cited by in corpus (10)

• Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo
• Variational methods for the kinetic Fokker-Planck equation
• Weighted $L^2$-contractivity of Langevin dynamics with singular potentials
• Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics
• Non-reversible lifts of reversible diffusion processes and relaxation times
• Complexity of zigzag sampling algorithm for strongly log-concave distributions
• Bosonized one-dimensional quantum systems through enhanced event-chain Monte Carlo
• Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration
• An entropic approach for Hamiltonian Monte Carlo: the idealized case
• Hypocoercivity meets lifts