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

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.