Hamiltonian Monte Carlo for Efficient Gaussian Sampling: Long and Random Steps

Abstract

Hamiltonian Monte Carlo (HMC) is a Markov chain algorithm for sampling from a high-dimensional distribution with density $e^{-f(x)}$, given access to the gradient of $f$. A particular case of interest is that of a $d$-dimensional Gaussian distribution with covariance matrix $\Sigma$, in which case $f(x) = x^\top \Sigma^{-1} x$. We show that Metropolis-adjusted HMC can sample from a distribution that is $\varepsilon$-close to a Gaussian in total variation distance using $\widetilde{O}(\sqrt{\kappa} d^{1/4} \log(1/\varepsilon))$ gradient queries, where $\varepsilon>0$ and $\kappa$ is the condition number of $\Sigma$. Our algorithm uses long and random integration times for the Hamiltonian dynamics, and it creates a warm start by first running HMC without a Metropolis adjustment. This contrasts with (and was motivated by) recent results that give an $\widetilde\Omega(\kappa d^{1/2})$ query lower bound for HMC with a fixed integration times or from a cold start, even for the Gaussian case.