Coupling and Convergence for Hamiltonian Monte Carlo

Abstract

Based on a new coupling approach, we prove that the transition step of the Hamiltonian Monte Carlo algorithm is contractive w.r.t. a carefully designed Kantorovich (L1 Wasserstein) distance. The lower bound for the contraction rate is explicit. Global convexity of the potential is not required, and thus multimodal target distributions are included. Explicit quantitative bounds for the number of steps required to approximate the stationary distribution up to a given error are a direct consequence of contractivity. These bounds show that HMC can overcome diffusive behaviour if the duration of the Hamiltonian dynamics is adjusted appropriately.

arXiv comment50 pages, 8 figures, extended the coupling approach to include corresponding results under a Foster-Lyapunov condition

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• Non-reversible lifts of reversible diffusion processes and relaxation times
• Mixing of Metropolis-Adjusted Markov Chains via Couplings: The High Acceptance Regime
• Complexity of zigzag sampling algorithm for strongly log-concave distributions
• Finding the jump rate for fastest decay in the Goldstein-Taylor model
• Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration
• Randomized Runge-Kutta-Nyström Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo
• Scalable couplings for the random walk Metropolis algorithm
• Estimating MCMC convergence rates using common random number simulation
• An entropic approach for Hamiltonian Monte Carlo: the idealized case