{
  "id": "2209.12467",
  "version": "v1",
  "published": "2022-09-26T07:16:50.000Z",
  "updated": "2022-09-26T07:16:50.000Z",
  "title": "Convergence rate of the (1+1)-evolution strategy on locally strongly convex functions with lipschitz continuous gradient and their monotonic transformations",
  "authors": [
    "Daiki Morinaga",
    "Kazuto Fukuchi",
    "Jun Sakuma",
    "Youhei Akimoto"
  ],
  "comment": "15 pages",
  "categories": [
    "math.OC",
    "cs.NE"
  ],
  "abstract": "Evolution strategy (ES) is one of promising classes of algorithms for black-box continuous optimization. Despite its broad successes in applications, theoretical analysis on the speed of its convergence is limited on convex quadratic functions and their monotonic transformation.%theoretically how fast it converges to a optima on convex functions is still vague. In this study, an upper bound and a lower bound of the rate of linear convergence of the (1+1)-ES on locally $L$-strongly convex functions with $U$-Lipschitz continuous gradient are derived as $\\exp\\left(-\\Omega_{d\\to\\infty}\\left(\\frac{L}{d\\cdot U}\\right)\\right)$ and $\\exp\\left(-\\frac1d\\right)$, respectively. Notably, any prior knowledge on the mathematical properties of the objective function such as Lipschitz constant is not given to the algorithm, whereas the existing analyses of derivative-free optimization algorithms require them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-26T07:16:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65K05",
      "90C25",
      "90C26",
      "90C56",
      "90C59",
      "G.1.6"
    ],
    "keywords": [
      "lipschitz continuous gradient",
      "locally strongly convex functions",
      "monotonic transformations",
      "convergence rate",
      "derivative-free optimization algorithms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}