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.

通过几何 Langevin MCMC 从一个 Riemann 流形 M 上的 Gibbs 分布 dπ* 进行高效采样的任务，我们提出了一种在实践中可实现的算法，该算法涉及在随机高斯方向上计算指数映射。通过对几何 Euler-Murayama 方案的离散化误差进行界定，假设▽h 是 Lipschitz 的且 M 具有有界的切向曲率，我们的误差界限与欧几里得 Euler-Murayama 的误差相匹配，结合 Kendall-Cranston 耦合下的几何 Langevin 扩散的收缩保证，我们证明 Langevin MCMC 迭代在经过～O (ε^-2) 次步骤后，与 π* 之间的 Wasserstein 距离小于 ε，这与欧几里得 Langevin MCMC 的迭代复杂性相匹配。我们的结果适用于具有非凸 h 和具有负 Ricci 曲率的一般设置。在额外的假设下，即 Riemann 曲率张量具有有界导数且 π* 满足 CD (・,∞) 条件，我们分析了 Langevin MCMC 的随机梯度版本，并将其迭代复杂性限制在～O (ε^-2) 次。