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 $\mathbb{R}<sup>{d}$, 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 $\mathcal{O}(d<sup>{\frac{1}{4}})$ gradient evaluations. This dependence on dimension is sharp, and our results significantly extend and improve previous quantitative bounds on the mixing of HMC.