{
  "abstract": "We prove the gap-entropy conjecture for fixed-confidence best-arm identification with independent unit-variance Gaussian arms, means in $[0,1]$, and a unique optimal arm. For each suboptimal arm $i$, let $Δ_i=μ_*-μ_i$ be its gap from the optimal mean, and write $H=\\sum_{i\\ne *}Δ_i^{-2}$. Let $p_r$ be the fraction of $H$ contributed by arms with $2^{-(r+1)}<Δ_i\\le2^{-r}$, and let $\\mathrm{Ent}(I)=\\sum_{r:p_r>0} p_r\\log(1/p_r)$. Among all algorithms that identify the optimal arm with probability at least $1-δ$ on every Gaussian instance, the optimal expected number of samples on a given instance, averaged over all permutations of the arm labels, is within absolute constant factors of $H(\\log(1/δ)+\\mathrm{Ent}(I))$. Moreover, there is an algorithm, independent of the instance, whose expected number of samples is bounded by a constant multiple of this quantity plus $g^{-2}\\log\\log(e^e/g)$, where $g=\\min_{i\\ne *}Δ_i$ is the gap to the closest competitor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10529",
  "author_total": 3,
  "authors": [
    {
      "name": "P. M. Aronow"
    },
    {
      "name": "Nathan Kallus"
    },
    {
      "name": "Patrick Lopatto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3c3p8p6b8z1xm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10529",
  "fulltext_url": null,
  "id": "3c3p8p6b8z1xm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10529",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1803|Advanced Bandit Algorithms Research"
    ]
  ],
  "openalex": "W7212114327",
  "pdf_url": "https://arxiv.org/pdf/2609.10529",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3c3p8p6b8z1xm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:22Z",
      "native_id": "2609.10529",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10529"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A positive resolution of the gap-entropy conjecture",
  "title_text": "A positive resolution of the gap-entropy conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:22Z",
  "url": "https://research.draconitic.com/paper/3c3p8p6b8z1xm",
  "venue": "arXiv",
  "versions": []
}