Hamiltonian Monte Carlo for efficient Gaussian sampling: long and random steps
(2209.12771)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 HMC can sample from a distribution that is $\varepsilon$-close in total variation distance using $\widetilde{O}(\sqrt{\kappa} d{1/4} \log(1/\varepsilon))$ gradient queries, where $\kappa$ is the condition number of $\Sigma$. Our algorithm uses long and random integration times for the Hamiltonian dynamics. This contrasts with (and was motivated by) recent results that give an $\widetilde\Omega(\kappa d{1/2})$ query lower bound for HMC with fixed integration times, even for the Gaussian case.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.