{
  "id": "2209.12771",
  "version": "v1",
  "published": "2022-09-26T15:29:29.000Z",
  "updated": "2022-09-26T15:29:29.000Z",
  "title": "Hamiltonian Monte Carlo for efficient Gaussian sampling: long and random steps",
  "authors": [
    "Simon Apers",
    "Sander Gribling",
    "Dániel Szilágyi"
  ],
  "comment": "21 pages",
  "categories": [
    "stat.ML",
    "cs.DS",
    "cs.LG"
  ],
  "abstract": "Hamiltonian Monte Carlo (HMC) is a Markov chain algorithm for sampling from a high-dimensional distribution with density $e^{-f(x)}$, given access to the gradient of $f$. A particular case of interest is that of a $d$-dimensional Gaussian distribution with covariance matrix $\\Sigma$, in which case $f(x) = x^\\top \\Sigma^{-1} x$. We show that HMC can sample from a distribution that is $\\varepsilon$-close in total variation distance using $\\widetilde{O}(\\sqrt{\\kappa} d^{1/4} \\log(1/\\varepsilon))$ gradient queries, where $\\kappa$ is the condition number of $\\Sigma$. Our algorithm uses long and random integration times for the Hamiltonian dynamics. This contrasts with (and was motivated by) recent results that give an $\\widetilde\\Omega(\\kappa d^{1/2})$ query lower bound for HMC with fixed integration times, even for the Gaussian case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-26T15:29:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamiltonian monte carlo",
      "efficient gaussian sampling",
      "random steps",
      "query lower bound",
      "dimensional gaussian distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}