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 M subset of R-n. We show that for distributions in the form of e(-alpha inverted perpendicular) (x) on a polytope with m constraints, the convergence rate of a family of commonly-used integrators is independent of ||alpha||(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 (O) over tilde (mn(3)) 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 Kook et al. (2022), which shows that RHMC with IMM can sample ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice.