Rapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions

TL;DR

This paper establishes that Hamiltonian Monte Carlo mixes rapidly for strongly log-concave distributions, providing dimension-free bounds and demonstrating efficient sampling with minimal gradient evaluations, thus advancing theoretical understanding of HMC's efficiency.

Contribution

The paper provides the first dimension-free bounds on HMC mixing for strongly log-concave distributions and shows that it can sample efficiently with only O(d^{1/4}) gradient evaluations.

Findings

HMC mixes rapidly for strongly log-concave distributions.

The mixing time bound is dimension-free.

Sampling requires only O(d^{1/4}) gradient evaluations.

Abstract

We obtain several quantitative bounds on the mixing properties of the Hamiltonian Monte Carlo (HMC) algorithm for a strongly log-concave target distribution $\pi$ on $\mathbb{R}^{d}$, showing that HMC mixes quickly in this setting. One of our main results is a dimension-free bound on the mixing of an "ideal" HMC chain, which is used to show that the usual leapfrog implementation of HMC can sample from $\pi$ using only $\mathcal{O}(d^{\frac{1}{4}})$ gradient evaluations. This dependence on dimension is sharp, and our results significantly extend and improve previous quantitative bounds on the mixing of HMC.