Sampling with Barriers: Faster Mixing via Lewis Weights

We analyze Riemannian Hamiltonian Monte Carlo (RHMC) for sampling a polytope defined by $m$ inequalities in $\R^n$ endowed with the metric defined by the Hessian of a self-concordant convex barrier function. We use a hybrid of the $p$-Lewis weight barrier and the standard logarithmic barrier and prove that the mixing rate is bounded by $\tilde O(m^{1/3}n^{4/3})$, improving on the previous best bound of $\tilde O(mn^{2/3})$, based on the log barrier. Our analysis overcomes several technical challenges to establish this result, in the process deriving smoothness bounds on Hamiltonian curves and extending self-concordance notions to the infinity norm; both properties appear to be of independent interest.