Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlo

Abstract

We show that the gradient norm $\|\nabla f(x)\|$ for $x \sim \exp(-f(x))$, where $f$ is strongly convex and smooth, concentrates tightly around its mean. This removes a barrier in the prior state-of-the-art analysis for the well-studied Metropolized Hamiltonian Monte Carlo (HMC) algorithm for sampling from a strongly logconcave distribution. We correspondingly demonstrate that Metropolized HMC mixes in $\tilde{O}(κd)$ iterations, improving upon the $\tilde{O}(κ^{1.5}\sqrt{d} + κd)$ runtime of (Dwivedi et. al. '18, Chen et. al. '19) by a factor $(κ/d)^{1/2}$ when the condition number $κ$ is large. Our mixing time analysis introduces several techniques which to our knowledge have not appeared in the literature and may be of independent interest, including restrictions to a nonconvex set with good conductance behavior, and a new reduction technique for boosting a constant-accuracy total variation guarantee under weak warmness assumptions. This is the first high-accuracy mixing time result for logconcave distributions using only first-order function information which achieves linear dependence on $κ$; we also give evidence that this dependence is likely to be necessary for standard Metropolized first-order methods.