Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation

TL;DR

This paper proves the convergence of Riemannian Hamiltonian Monte Carlo and applies it to efficiently sample polytopes and compute their volume, achieving improved step complexity bounds.

Contribution

It provides the first rigorous convergence proof for Riemannian Hamiltonian Monte Carlo and extends Gaussian cooling to manifolds for volume computation.

Findings

Convergence rate bounded by smoothness parameters of the manifold.

Achieves O^{*}(mn^{2/3}) steps for sampling and volume estimation.

Extends Gaussian cooling to Riemannian manifolds.

Abstract

We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of convergence is bounded in terms of natural smoothness parameters of an associated Riemannian manifold. We then apply the method with the manifold defined by the log barrier function to the problems of (1) uniformly sampling a polytope and (2) computing its volume, the latter by extending Gaussian cooling to the manifold setting. In both cases, the total number of steps needed is O^{*}(mn^{\frac{2}{3}}), improving the state of the art. A key ingredient of our analysis is a proof of an analog of the KLS conjecture for Gibbs distributions over manifolds.