Condition-number-independent Convergence Rate of Riemannian Hamiltonian Monte Carlo with Numerical Integrators

• URL: http://arxiv.org/abs/2210.07219v1
• Date: Thu, 13 Oct 2022 17:46:51 GMT
• Title: Condition-number-independent Convergence Rate of Riemannian Hamiltonian Monte Carlo with Numerical Integrators
• Authors: Yunbum Kook, Yin Tat Lee, Ruoqi Shen, Santosh S. Vempala
• Abstract summary: We show that for distributions in the form of $e-alphatopx$ on a polytope with $m constraints, the convergence rate of a family of commonly-used$$ is independent of $leftVert alpharightVert$ and the geometry of the polytope. 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.
• Score: 22.49731518828916
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 set $\mathcal{M}\subset\mathbb{R}^{n}$. 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 integrator (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.

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