Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo

TL;DR

This paper establishes quantitative bounds on the mixing time of unadjusted Hamiltonian Monte Carlo, showing it depends logarithmically on dimension and providing practical gradient evaluation estimates for accurate sampling.

Contribution

It offers the first explicit upper bounds on the mixing time of uHMC and quantifies its approximation accuracy for broad classes of models.

Findings

Mixing time depends logarithmically on dimension.

Gradient evaluations needed scale as $O(d^{3/4} ext{or }d^{1/2} ext{ with }\varepsilon)$.

uHMC can approximate the target distribution within specified total variation distance.

Abstract

We provide quantitative upper bounds on the total variation mixing time of the Markov chain corresponding to the unadjusted Hamiltonian Monte Carlo (uHMC) algorithm. For two general classes of models and fixed time discretization step size $h$, the mixing time is shown to depend only logarithmically on the dimension. Moreover, we provide quantitative upper bounds on the total variation distance between the invariant measure of the uHMC chain and the true target measure. As a consequence, we show that an $\varepsilon$-accurate approximation of the target distribution $\mu$ in total variation distance can be achieved by uHMC for a broad class of models with $O\left(d^{3/4}\varepsilon^{-1/2}\log (d/\varepsilon )\right)$ gradient evaluations, and for mean field models with weak interactions with $O\left(d^{1/2}\varepsilon^{-1/2}\log (d/\varepsilon )\right)$ gradient evaluations. The proofs are based on the construction of successful couplings for uHMC that realize the upper bounds.