Efficient Sampling on Riemannian Manifolds via Langevin MCMC

TL;DR

This paper develops an efficient Langevin MCMC algorithm for sampling from Gibbs distributions on Riemannian manifolds, providing theoretical guarantees on convergence and complexity that extend Euclidean results to curved spaces.

Contribution

It introduces a geometric Langevin MCMC method with error bounds and convergence guarantees applicable to general Riemannian manifolds, including nonconvex and negatively curved cases.

Findings

Langevin MCMC converges within $ ilde{O}( ext{epsilon}^{-2})$ steps.

Error bounds match Euclidean case under Lipschitz gradient assumptions.

Applicable to nonconvex and negatively curved manifolds.

Abstract

We study the task of efficiently sampling from a Gibbs distribution $d \pi^* = e^{-h} d {vol}_g$ over a Riemannian manifold $M$ via (geometric) Langevin MCMC; this algorithm involves computing exponential maps in random Gaussian directions and is efficiently implementable in practice. The key to our analysis of Langevin MCMC is a bound on the discretization error of the geometric Euler-Murayama scheme, assuming $\nabla h$ is Lipschitz and $M$ has bounded sectional curvature. Our error bound matches the error of Euclidean Euler-Murayama in terms of its stepsize dependence. Combined with a contraction guarantee for the geometric Langevin Diffusion under Kendall-Cranston coupling, we prove that the Langevin MCMC iterates lie within $\epsilon$-Wasserstein distance of $\pi^*$ after $\tilde{O}(\epsilon^{-2})$ steps, which matches the iteration complexity for Euclidean Langevin MCMC. Our results apply in general settings where $h$ can be nonconvex and $M$ can have negative Ricci curvature. Under additional assumptions that the Riemannian curvature tensor has bounded derivatives, and that $\pi^*$ satisfies a $CD(\cdot,\infty)$ condition, we analyze the stochastic gradient version of Langevin MCMC, and bound its iteration complexity by $\tilde{O}(\epsilon^{-2})$ as well.