Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlo

• URL: http://arxiv.org/abs/2002.04121v3
• Date: Sun, 14 Jun 2020 02:12:45 GMT
• Title: Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlo
• Authors: Yin Tat Lee, Ruoqi Shen, Kevin Tian
• Abstract summary: This is the first high-accuracy mixing time result for logconcave distributions using only first-order function information. We give evidence that dependence on $kappa$ is likely to be necessary standard Metropolized firstorder methods.
• Score: 23.781520510778716
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\tilde{O}(\kappa d)$ iterations, improving upon the $\tilde{O}(\kappa^{1.5}\sqrt{d} + \kappa d)$ runtime of (Dwivedi et. al. '18, Chen et. al. '19) by a factor $(\kappa/d)^{1/2}$ when the condition number $\kappa$ 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 $\kappa$; we also give evidence that this dependence is likely to be necessary for standard Metropolized first-order methods.

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