Efficient Sampling on Riemannian Manifolds via Langevin MCMC

We study the task of efficiently sampling from a Gibbs distribution d π^* = 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 ∇ 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 ϵ-Wasserstein distance of π^* after Õ(ϵ^-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 π^* satisfies a CD(·,∞) condition, we analyze the stochastic gradient version of Langevin MCMC, and bound its iteration complexity by Õ(ϵ^-2) as well.