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

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 ℳ⊂ℝ^n. We show that for distributions in the form of e^-α^⊤x on a polytope with m constraints, the convergence rate of a family of commonly-used integrators is independent of ‖α‖_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 O(mn^3) to achieve ϵ 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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