{
  "id": "1408.4622",
  "version": "v1",
  "published": "2014-08-20T12:16:54.000Z",
  "updated": "2014-08-20T12:16:54.000Z",
  "title": "A new integral loss function for Bayesian optimization",
  "authors": [
    "Emmanuel Vazquez",
    "Julien Bect"
  ],
  "comment": "6 pages",
  "categories": [
    "stat.CO",
    "cs.LG",
    "math.OC",
    "stat.ML"
  ],
  "abstract": "We consider the problem of maximizing a real-valued continuous function $f$ using a Bayesian approach. Since the early work of Jonas Mockus and Antanas \\v{Z}ilinskas in the 70's, the problem of optimization is usually formulated by considering the loss function $\\max f - M_n$ (where $M_n$ denotes the best function value observed after $n$ evaluations of $f$). This loss function puts emphasis on the value of the maximum, at the expense of the location of the maximizer. In the special case of a one-step Bayes-optimal strategy, it leads to the classical Expected Improvement (EI) sampling criterion. This is a special case of a Stepwise Uncertainty Reduction (SUR) strategy, where the risk associated to a certain uncertainty measure (here, the expected loss) on the quantity of interest is minimized at each step of the algorithm. In this article, assuming that $f$ is defined over a measure space $(\\mathbb{X}, \\lambda)$, we propose to consider instead the integral loss function $\\int_{\\mathbb{X}} (f - M_n)_{+}\\, d\\lambda$, and we show that this leads, in the case of a Gaussian process prior, to a new numerically tractable sampling criterion that we call $\\rm EI^2$ (for Expected Integrated Expected Improvement). A numerical experiment illustrates that a SUR strategy based on this new sampling criterion reduces the error on both the value and the location of the maximizer faster than the EI-based strategy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-20T12:16:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integral loss function",
      "bayesian optimization",
      "sampling criterion",
      "special case",
      "one-step bayes-optimal strategy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.4622V"
    }
  }
}