{
  "abstract": "We study the sampling allocation of LinUCB in the small-gap regime, where the reward gaps are of order at most $n^{-1/2}$ over the decision horizon $n$. This scaling captures the hard instances underlying worst-case regret lower bounds, for which LinUCB is known to be near optimal up to logarithmic factors in $n$. Using a mean-field perspective, we characterize this allocation through the empirical sampling distribution, a macroscopic object that averages the effect of adaptive decisions over the horizon, and identify its limit as $n\\to\\infty$. We establish that in this regime, the empirical sampling distribution induced by LinUCB converges to the set of D-optimal designs. This central result reveals that, in the small-gap regime, LinUCB not only achieves near optimal minimax regret but also allocates samples in a way that is asymptotically efficient for learning the reward parameter, thereby connecting regret-driven online learning with information-efficient experimental design. Building on the optimal design limit, we obtain two useful consequences. First, we refine the asymptotic regret analysis of LinUCB in the small-gap regime by characterizing its leading-order constant in the limit. Second, we show that, despite LinUCB's adaptive sampling strategy, the regularized least-squares estimator satisfies a central-limit-type theorem in the small-gap regime, thereby enabling valid statistical inference for the reward parameter.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06213",
  "author_total": 3,
  "authors": [
    {
      "name": "Yujie Liu"
    },
    {
      "name": "Vincent Y. F. Tan"
    },
    {
      "name": "Yunbei Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/575tja859zx0c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06213",
  "fulltext_url": null,
  "id": "575tja859zx0c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06213",
  "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": "W7220669937",
  "pdf_url": "https://arxiv.org/pdf/2610.06213",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/575tja859zx0c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:16Z",
      "native_id": "2610.06213",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06213"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Sampling Allocation of LinUCB: Optimal Design Limits in the Small-Gap Regime",
  "title_text": "Sampling Allocation of LinUCB: Optimal Design Limits in the Small-Gap Regime",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/575tja859zx0c",
  "venue": "arXiv",
  "versions": []
}