1 research outputs found
Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients
Hamiltonian Monte Carlo (HMC) is a state-of-the-art Markov chain Monte Carlo
sampling algorithm for drawing samples from smooth probability densities over
continuous spaces. We study the variant most widely used in practice,
Metropolized HMC with the St\"{o}rmer-Verlet or leapfrog integrator, and make
two primary contributions. First, we provide a non-asymptotic upper bound on
the mixing time of the Metropolized HMC with explicit choices of step-size and
number of leapfrog steps. This bound gives a precise quantification of the
faster convergence of Metropolized HMC relative to simpler MCMC algorithms such
as the Metropolized random walk, or Metropolized Langevin algorithm. Second, we
provide a general framework for sharpening mixing time bounds of Markov chains
initialized at a substantial distance from the target distribution over
continuous spaces. We apply this sharpening device to the Metropolized random
walk and Langevin algorithms, thereby obtaining improved mixing time bounds
from a non-warm initial distribution.Comment: 73 pages, 2 figures, fixed a mistake in the proof of Lemma 11,
accepted in JML