We study the convergence rate of discretized Riemannian Hamiltonian Monte Carlo on sampling from distributions in the form of $e^{-f(x)}$ on a convex body $\mathcal{M}\subset\mathbb{R}^{n}$. We show that for distributions in the form of $e^{-\alpha^{\top}x}$ on a polytope with $m$ constraints, the convergence rate of a family of commonly-used integrators is independent of $\left\Vert \alpha\right\Vert _{2}$ and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of $\widetilde{O}\left(mn^{3}\right)$ to achieve $\epsilon$ total variation distance to the target distribution. These guarantees are based on a general bound on the convergence rate for densities of the form $e^{-f(x)}$ in terms of parameters of the manifold and the integrator. Our theoretical guarantee complements the empirical results of [KLSV22], which shows that RHMC with IMM can sample ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice.

翻译：我们研究了离散化黎曼哈密顿蒙特卡洛方法在凸体 $\mathcal{M}\subset\mathbb{R}^{n}$ 上采样形如 $e^{-f(x)}$ 分布时的收敛率。结果表明，对于具有 $m$ 个约束的多面体上形如 $e^{-\alpha^{\top}x}$ 的分布，一类常用积分器的收敛率独立于 $\left\Vert \alpha\right\Vert _{2}$ 及多面体的几何特性。特别地，隐式中点法（IMM）和广义蛙跳法（LM）在达到与目标分布总变差距离 $\epsilon$ 时，混合时间仅为 $\widetilde{O}\left(mn^{3}\right)$。这些保证基于形如 $e^{-f(x)}$ 密度函数的收敛率关于流形与积分器参数的通用上界。我们的理论证明补充了 [KLSV22] 的实证结果——该工作表明采用 IMM 的 RHMC 可在极高维空间中高效采样病态、非光滑且带约束的分布。