{
  "abstract": "We study the Lipschitz bandit problem, where a learner sequentially maximizes an unknown Lipschitz function $f$ over a domain $\\mathcal{X} \\subset [0,1]^d$ using noisy pointwise evaluations. Existing regret bounds are either worst-case, scaling as $\\tildeΘ \\left ( T^{d+1/d+2}\\right )$, or adaptive via the zooming dimension $d_z$, yielding $\\tildeΘ \\left ( T^{d_z+1/d_z+2}\\right )$. However, such zooming-based guarantees are only partially instance-dependent, as they depend solely on the asymptotic growth of near-optimal level sets and fail to capture finer structural properties of $f$. We provide an analysis and an algorithm that characterizes the regret through integrals of the suboptimality gap of $f$ over its level sets. This yields regret bounds that adapt to the local growth of level sets, rather than only their asymptotic behavior. As a corollary, when the set of maximizers has dimension $d^\\star>0$, we obtain improved adaptive rates of order $\\tilde{\\mathcal{O}} \\left ( T^{d_z+1 / \\max(d_z,d^\\star)+2}\\right )$ strictly improving over classical zooming bounds in this regime. Finally, we extend our analysis to the full-information setting (Lipschitz experts) and show how some of the regularity assumptions can be relaxed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.29748",
  "author_total": 2,
  "authors": [
    {
      "name": "Marius Potfer"
    },
    {
      "name": "Vianney Perchet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cd00b5p6qff9n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.29748",
  "fulltext_url": null,
  "id": "cd00b5p6qff9n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.29748",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1803|Advanced Bandit Algorithms Research"
    ]
  ],
  "openalex": "W7162780981",
  "pdf_url": "https://arxiv.org/pdf/2605.29748",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cd00b5p6qff9n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:25Z",
      "native_id": "2605.29748",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.29748"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Instance-dependent Stochastic Lipschitz bandit",
  "title_text": "Instance-dependent Stochastic Lipschitz bandit",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:25Z",
  "url": "https://research.draconitic.com/paper/cd00b5p6qff9n",
  "venue": "arXiv",
  "versions": []
}