Accelerating Hamiltonian Monte Carlo via Chebyshev Integration Time

TL;DR

This paper introduces a Chebyshev polynomial-based scheme for time-varying integration time in Hamiltonian Monte Carlo, achieving accelerated convergence from linear to square-root dependence on the condition number for Gaussian targets and showing benefits for more general distributions.

Contribution

It proposes a novel Chebyshev polynomial-based method for adaptive integration time in HMC, leading to accelerated sampling efficiency especially for Gaussian targets.

Findings

Achieves $O(\sqrt{\kappa} \log \frac{1}{\epsilon})$ iteration complexity for Gaussian targets.

Demonstrates improved convergence over constant integration time schemes.

Experimental results show advantages for non-quadratic strongly convex potentials.

Abstract

Hamiltonian Monte Carlo (HMC) is a popular method in sampling. While there are quite a few works of studying this method on various aspects, an interesting question is how to choose its integration time to achieve acceleration. In this work, we consider accelerating the process of sampling from a distribution $\pi(x) \propto \exp(-f(x))$ via HMC via time-varying integration time. When the potential $f$ is $L$-smooth and $m$-strongly convex, i.e.\ for sampling from a log-smooth and strongly log-concave target distribution $\pi$, it is known that under a constant integration time, the number of iterations that ideal HMC takes to get an $\epsilon$ Wasserstein-2 distance to the target $\pi$ is $O( \kappa \log \frac{1}{\epsilon} )$, where $\kappa := \frac{L}{m}$ is the condition number. We propose a scheme of time-varying integration time based on the roots of Chebyshev polynomials. We show that in the case of quadratic potential $f$, i.e., when the target $\pi$ is a Gaussian distribution, ideal HMC with this choice of integration time only takes $O( \sqrt{\kappa} \log \frac{1}{\epsilon} )$ number of iterations to reach Wasserstein-2 distance less than $\epsilon$; this improvement on the dependence on condition number is akin to acceleration in optimization. The design and analysis of HMC with the proposed integration time is built on the tools of Chebyshev polynomials. Experiments find the advantage of adopting our scheme of time-varying integration time even for sampling from distributions with smooth strongly convex potentials that are not quadratic.