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 body Msubsetℝn. We show that for distributions in the form of e^-αtopx on a polytope with m constraints, the convergence rate of a family of commonly-used integrators is independent of leftVert αrightVert 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 widetildeOleft(mn3right) 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.