{
  "id": "2102.12956",
  "version": "v1",
  "published": "2021-02-25T16:03:04.000Z",
  "updated": "2021-02-25T16:03:04.000Z",
  "title": "Stein Variational Gradient Descent: many-particle and long-time asymptotics",
  "authors": [
    "Nikolas Nüsken",
    "D. R. Michiel Renger"
  ],
  "comment": "25 pages",
  "categories": [
    "stat.ML",
    "cs.LG",
    "cs.NA",
    "math.AP",
    "math.NA",
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: variational inference and Markov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the Stein-Fisher information (or kernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $\\Gamma$-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-25T16:03:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stein variational gradient descent",
      "long-time asymptotics",
      "markov chain monte carlo",
      "stein-fisher information",
      "bayesian computational statistics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}