Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions

TL;DR

This paper establishes explicit, dimension-free convergence bounds for the preconditioned Hamiltonian Monte Carlo algorithm in infinite-dimensional spaces, applicable to high-dimensional sampling problems without requiring convex potentials.

Contribution

It introduces a novel two-scale coupling method that proves contractivity and provides non-asymptotic convergence bounds for pHMC in Hilbert spaces.

Findings

Dimension-free convergence bounds derived

Applicable to high-dimensional transition path sampling

No global convexity assumption needed

Abstract

We derive non-asymptotic quantitative bounds for convergence to equilibrium of the exact preconditioned Hamiltonian Monte Carlo algorithm (pHMC) on a Hilbert space. As a consequence, explicit and dimension-free bounds for pHMC applied to high-dimensional distributions arising in transition path sampling and path integral molecular dynamics are given. Global convexity of the underlying potential energies is not required. Our results are based on a two-scale coupling which is contractive in a carefully designed distance.