Efficient Sampling on Riemannian Manifolds via Langevin MCMC

• URL: http://arxiv.org/abs/2402.10357v1
• Date: Thu, 15 Feb 2024 22:59:14 GMT
• Title: Efficient Sampling on Riemannian Manifolds via Langevin MCMC
• Authors: Xiang Cheng, Jingzhao Zhang, Suvrit Sra
• Abstract summary: We study the task efficiently sampling from a Gibbs distribution $d pi* = eh d vol_g$ over aian manifold $M$ via (geometric) Langevin MCMC. Our results apply to general settings where $pi*$ can be non exponential and $Mh$ can have negative Ricci curvature.
• Score: 51.825900634131486
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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