We analyze Riemannian Hamiltonian Monte Carlo (RHMC) for sampling a polytope defined by $m$ inequalities in $\R^n$ endowed with the metric defined by the Hessian of a convex barrier function. The advantage of RHMC over Euclidean methods such as the ball walk, hit-and-run and the Dikin walk is in its ability to take longer steps. However, in all previous work, the mixing rate has a linear dependence on the number of inequalities. We introduce a hybrid of the Lewis weights barrier and the standard logarithmic barrier and prove that the mixing rate for the corresponding RHMC is bounded by $\tilde O(m^{1/3}n^{4/3})$, improving on the previous best bound of $\tilde O(mn^{2/3})$ (based on the log barrier). This continues the general parallels between optimization and sampling, with the latter typically leading to new tools and more refined analysis. To prove our main results, we have to overcomes several challenges relating to the smoothness of Hamiltonian curves and the self-concordance properties of the barrier. In the process, we give a general framework for the analysis of Markov chains on Riemannian manifolds, derive new smoothness bounds on Hamiltonian curves, a central topic of comparison geometry, and extend self-concordance to the infinity norm, which gives sharper bounds; these properties appear to be of independent interest.

翻译：我们分析了黎曼哈密尔顿蒙特卡洛（RHMC）方法，用于采样由 $\R^n$ 中 $m$ 个不等式定义的多面体，该空间配备了由凸障碍函数的海森矩阵定义的度量。RHMC 相较于欧几里得方法（如球面行走、击跑行走和迪金行走）的优势在于其能够采取更长的步长。然而，在以往的所有工作中，混合速率对不等式数量呈线性依赖。我们引入了一种结合刘易斯权重障碍与标准对数障碍的混合方法，并证明相应 RHMC 的混合速率受限于 $\tilde O(m^{1/3}n^{4/3})$，优于此前基于对数障碍的最佳界 $\tilde O(mn^{2/3})$。这延续了优化与采样之间的普遍平行关系，其中后者通常催生新工具和更精细的分析。为证明主要结果，我们需要克服若干与哈密尔顿曲线光滑性及障碍自和谐性质相关的挑战。在此过程中，我们提出一个用于分析黎曼流形上马尔可夫链的通用框架，推导出哈密尔顿曲线（比较几何的核心主题）的新光滑界，并将自和谐性扩展到无穷范数以获得更紧的界；这些性质似乎具有独立的研究价值。