{
  "id": "1404.0105",
  "version": "v3",
  "published": "2014-04-01T02:05:36.000Z",
  "updated": "2014-08-31T23:13:22.000Z",
  "title": "Irreversible Langevin samplers and variance reduction: a large deviation approach",
  "authors": [
    "Luc Rey-Bellet",
    "Kostantinos Spiliopoulos"
  ],
  "comment": "21 pages, 4 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In order to sample from a given target distribution (often of Gibbs type), the Monte Carlo Markov chain method consists in constructing an ergodic Markov process whose invariant measure is the target distribution. By sampling the Markov process one can then compute, approximately, expectations of observables with respect to the target distribution. Often the Markov processes used in practice are time-reversible (i.e., they satisfy detailed balance), but our main goal here is to assess and quantify how the addition of a non-reversible part to the process can be used to improve the sampling properties. We focus on the diffusion setting (overdamped Langevin equations) where the drift consists of a gradient vector field as well as another drift which breaks the reversibility of the process but is chosen to preserve the Gibbs measure. In this paper we use the large deviation rate function for the empirical measure as a tool to analyze the speed of convergence to the invariant measure. We show that the addition of an irreversible drift leads to a larger rate function and it strictly improves the speed of convergence of ergodic average for (generic smooth) observables. We also deduce from this result that the asymptotic variance decreases under the addition of the irreversible drift and we give an explicit characterization of the observables whose do not see their variances reduced, in terms of a nonlinear Poisson equation. Our theoretical results are illustrated and supplemented by numerical simulations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-03T19:09:08.000Z",
      "abstract": "Monte Carlo methods are a popular tool to sample from high-dimensional target distribution and to approximate expectations of observables with respect to a given distribution, which very often is of Gibbs type. Sampling an ergodic Markov process which has the target distribution as its invariant measure can be used to compute approximately such expectations. Often the Markov processes used are time-reversible (i.e., they satisfy detailed balance) but our main goal here is to assess and quantify, in a novel way, how adding an irreversible part to the process can be used to improve the sampling properties. We focus on the diffusion setting (overdamped Langevin equations) and we explore performance criteria based on the large deviations theory for empirical measures. We find that large deviations theory can not only adequately characterize the efficiency of the approximations, but it can also be used as a vehicle to design Markov processes, whose time averages optimally (in the sense of variance reduction) approximates the quantities of interest. We quantify the effect of the added irreversibility on the speed of convergence to the target Gibbs measure and to the asymptotic variance of the resulting estimators for observables. One of our main finding is that adding irreversibility reduces the asymptotic variance of generic observables and we give an explicit characterization of when observables do not see their variances reduced in terms of a nonlinear Poisson equation. Our theoretical results are illustrated and supplemented by numerical simulations.",
      "comment": "23 pages, 6 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-08-31T23:13:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60F10",
      "60J25",
      "60J60",
      "65C05",
      "82B80"
    ],
    "keywords": [
      "large deviation approach",
      "irreversible langevin samplers",
      "variance reduction",
      "target distribution",
      "monte carlo markov chain method"
    ],
    "publication": {
      "doi": "10.1088/0951-7715/28/7/2081",
      "journal": "Nonlinearity",
      "year": 2015,
      "month": "Jul",
      "volume": 28,
      "number": 7,
      "pages": 2081
    },
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015Nonli..28.2081R"
    }
  }
}