{
  "id": "2302.03973",
  "version": "v1",
  "published": "2023-02-08T10:08:37.000Z",
  "updated": "2023-02-08T10:08:37.000Z",
  "title": "Improved Langevin Monte Carlo for stochastic optimization via landscape modification",
  "authors": [
    "Michael C. H. Choi",
    "Youjia Wang"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR",
    "cs.LG",
    "math.OC",
    "stat.CO",
    "stat.ML"
  ],
  "abstract": "Given a target function $H$ to minimize or a target Gibbs distribution $\\pi_{\\beta}^0 \\propto e^{-\\beta H}$ to sample from in the low temperature, in this paper we propose and analyze Langevin Monte Carlo (LMC) algorithms that run on an alternative landscape as specified by $H^f_{\\beta,c,1}$ and target a modified Gibbs distribution $\\pi^f_{\\beta,c,1} \\propto e^{-\\beta H^f_{\\beta,c,1}}$, where the landscape of $H^f_{\\beta,c,1}$ is a transformed version of that of $H$ which depends on the parameters $f,\\beta$ and $c$. While the original Log-Sobolev constant affiliated with $\\pi^0_{\\beta}$ exhibits exponential dependence on both $\\beta$ and the energy barrier $M$ in the low temperature regime, with appropriate tuning of these parameters and subject to assumptions on $H$, we prove that the energy barrier of the transformed landscape is reduced which consequently leads to polynomial dependence on both $\\beta$ and $M$ in the modified Log-Sobolev constant associated with $\\pi^f_{\\beta,c,1}$. This yield improved total variation mixing time bounds and improved convergence toward a global minimum of $H$. We stress that the technique developed in this paper is not only limited to LMC and is broadly applicable to other gradient-based optimization or sampling algorithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-08T10:08:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic optimization",
      "landscape modification",
      "total variation mixing time bounds",
      "energy barrier",
      "gibbs distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}