Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients

TL;DR

This paper provides explicit mixing time bounds for Metropolized Hamiltonian Monte Carlo, demonstrating its faster convergence compared to simpler algorithms, and introduces a framework to improve bounds from distant initializations.

Contribution

It offers the first non-asymptotic mixing time bounds for Metropolized HMC with explicit parameters and develops a framework to sharpen bounds for chains starting far from the target.

Findings

Metropolized HMC converges faster than random walk and Langevin algorithms.

Explicit mixing time bounds are derived for practical step-size and leapfrog steps.

A new framework improves mixing time bounds from distant initial distributions.

Abstract

Hamiltonian Monte Carlo (HMC) is a state-of-the-art Markov chain Monte Carlo sampling algorithm for drawing samples from smooth probability densities over continuous spaces. We study the variant most widely used in practice, Metropolized HMC with the St\"{o}rmer-Verlet or leapfrog integrator, and make two primary contributions. First, we provide a non-asymptotic upper bound on the mixing time of the Metropolized HMC with explicit choices of step-size and number of leapfrog steps. This bound gives a precise quantification of the faster convergence of Metropolized HMC relative to simpler MCMC algorithms such as the Metropolized random walk, or Metropolized Langevin algorithm. Second, we provide a general framework for sharpening mixing time bounds of Markov chains initialized at a substantial distance from the target distribution over continuous spaces. We apply this sharpening device to the Metropolized random walk and Langevin algorithms, thereby obtaining improved mixing time bounds from a non-warm initial distribution.