{
  "id": "2305.07241",
  "version": "v1",
  "published": "2023-05-12T04:12:12.000Z",
  "updated": "2023-05-12T04:12:12.000Z",
  "title": "On the Optimality of Misspecified Kernel Ridge Regression",
  "authors": [
    "Haobo Zhang",
    "Yicheng Li",
    "Weihao Lu",
    "Qian Lin"
  ],
  "comment": "23 pages, 6 figures, The Fortieth International Conference on Machine Learning. arXiv admin note: substantial text overlap with arXiv:2303.14942",
  "categories": [
    "cs.LG",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "In the misspecified kernel ridge regression problem, researchers usually assume the underground true function $f_{\\rho}^{*} \\in [\\mathcal{H}]^{s}$, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) $\\mathcal{H}$ for some $s\\in (0,1)$. The existing minimax optimal results require $\\|f_{\\rho}^{*}\\|_{L^{\\infty}}<\\infty$ which implicitly requires $s > \\alpha_{0}$ where $\\alpha_{0}\\in (0,1)$ is the embedding index, a constant depending on $\\mathcal{H}$. Whether the KRR is optimal for all $s\\in (0,1)$ is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any $s\\in (0,1)$ when the $\\mathcal{H}$ is a Sobolev RKHS.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-12T04:12:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "misspecified kernel ridge regression problem",
      "optimality",
      "underground true function",
      "existing minimax optimal results",
      "less-smooth interpolation space"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}