Convergence of Preconditioned Hamiltonian Monte Carlo on Hilbert Spaces

TL;DR

This paper establishes convergence bounds for the preconditioned Hamiltonian Monte Carlo algorithm in infinite-dimensional Hilbert spaces, under conditions similar to strong log-concavity, using coupling techniques.

Contribution

It extends convergence analysis of pHMC to infinite-dimensional spaces, providing theoretical guarantees under specific conditions.

Findings

Proves convergence bounds in 1-Wasserstein distance

Uses coupling methods for analysis

Applicable to infinite-dimensional Hilbert spaces

Abstract

In this article, we consider the preconditioned Hamiltonian Monte Carlo (pHMC) algorithm defined directly on an infinite-dimensional Hilbert space. In this context, and under a condition reminiscent of strong log-concavity of the target measure, we prove convergence bounds for adjusted pHMC in the standard 1-Wasserstein distance. The arguments rely on a synchronous coupling of two copies of pHMC, which is controlled by adapting elements from arXiv:1805.00452.