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:我们研究了分解的里曼尼安·汉密尔顿·蒙特卡洛在以美元-f(x)美元的形式对一个复合体的分布采样进行采样的混合率。我们研究的是,对于以美元为限制条件的聚点以美元的形式分配的里曼尼尼安·汉密尔顿·蒙特卡洛,一个常用集成商家庭的合并率独立于$left\Vert\alpha\right\Vert ⁇ 2}美元和多元体的几何学。特别是,隐含的中点方法(IMM)和通用的Leapmorog 方法(LM)的混合时间是$\blitede{O ⁇ left(m ⁇ 3 ⁇ right)美元,以达到美元总与目标分布的距离。这些保证的基础是,以美元-f(x)美元形式的密度的合并率为基础,从参数上看,隐含的中点方法(IMMMM-22)法方法(LMR)的混合时间段间混合时间,以美元为美元,以高度的模型和高比例的模型分析结果补充了我们的RIS-结果。