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 $\mathcal{M}\subset {\mathbb{R}}^n$. We show that for distributions in the form of $e^{-{\alpha }^{\mathrm{\top }}x}$ on a polytope with $m$ constraints, the convergence rate of a family of commonly-used integrators is independent of ${\parallel \alpha \parallel }_2$ and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of $\tilde{O}\left(m{n}^3\right)$ 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.