{
  "abstract": "We study expected improvement (EI) for minimizing a deterministic function $f$ in the RKHS $\\mathcal H_k$ of a continuous positive-semidefinite kernel $k$ on a nonempty compact set $\\mathcal X\\subset\\mathbb R^d$. Function values are observed exactly, and EI is computed from a fixed zero-mean Gaussian-process model with covariance $σ^2k$, $σ>0$. A weak-EI policy queries a point whose EI is at least a fixed positive fraction of its maximum. We introduce a notion of sequential separation radius relating ranked selected-point innovation norms to Kolmogorov widths, drawing on greedy approximation. Standard power-function estimates from scattered-data approximation and a finite-budget regret argument yield the rates. After $N$ post-initial queries, every weak-EI policy has simple regret $O(N^{-ν/d})$ for isotropic Matérn kernels of smoothness $ν>0$ and $O(\\exp[-c_1\\min\\{N,N^{1/d}\\log(eN)\\}])$ for the isotropic squared-exponential kernel, with $c_1>0$. For $d=1$, the sharper bound $O(\\exp[-c_2N\\log(eN)])$ holds for exact EI, with $c_2>0$. These bounds are uniform over each fixed RKHS ball. If $\\mathcal X$ has nonempty interior and $B>0$, the exact EI policy is minimax-rate optimal over the RKHS ball of radius $B$ for Matérn kernels, even among randomized strategies whose final recommendation need not be a query point. For the squared-exponential kernel, it is minimax-rate optimal up to constants in the exponent among deterministic methods whose final recommendation may be any point of $\\mathcal X$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.29245",
  "author_total": 2,
  "authors": [
    {
      "name": "Emmanuel Vazquez"
    },
    {
      "name": "Sébastien Petit"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8fqwgf8sm6ajf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.29245",
  "fulltext_url": null,
  "id": "8fqwgf8sm6ajf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.29245",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1803|Advanced Bandit Algorithms Research"
    ]
  ],
  "openalex": "W7172296957",
  "pdf_url": "https://arxiv.org/pdf/2607.29245",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8fqwgf8sm6ajf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2607.29245",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.29245"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Simple-regret rates and minimax optimality of fixed-prior expected improvement in Matérn and squared-exponential RKHSs",
  "title_text": "Simple-regret rates and minimax optimality of fixed-prior expected improvement in Matérn and squared-exponential RKHSs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:10Z",
  "url": "https://research.draconitic.com/paper/8fqwgf8sm6ajf",
  "venue": "arXiv",
  "versions": []
}