Coupling and Convergence for Hamiltonian Monte Carlo

TL;DR

This paper introduces a new coupling approach to prove that Hamiltonian Monte Carlo (HMC) exhibits contractive behavior under a specific Wasserstein distance, providing explicit bounds on convergence rates even for multimodal distributions.

Contribution

It presents a novel coupling method demonstrating HMC's contractivity without requiring convex potentials, with explicit bounds on convergence time.

Findings

HMC is contractive under a Wasserstein distance.

Explicit bounds on the number of steps for convergence.

HMC can overcome diffusive behavior with proper tuning.

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.