{
  "abstract": "We study fair multi-armed bandits under the Nash Social Welfare (NSW) objective, which measures performance via the geometric mean of accumulated rewards. Existing work defines Nash regret as $\\mathrm{NR}_T = μ^\\star - (\\prod_{t=1}^T \\mathbb{E}μ_{I_t})^{1/T}$, where $μ_{I_t}$ is the mean reward of the recommended arm $I_t$ and $T$ is the horizon. Since it applies the geometric mean to per-round marginal expectations, it ignores the joint distribution of rewards across rounds, leaving the NSW fairness motivation unaddressed at the trajectory level. We propose \\emph{trajectory-wise Nash regret} $\\widetilde{\\mathrm{NR}}_T = μ^\\star - \\mathbb{E}[(\\prod_{t=1}^T μ_{I_t})^{1/T}]$, which computes the geometric mean over complete sample paths before taking expectations, capturing NSW fairness more faithfully. By Jensen's inequality, $\\widetilde{\\mathrm{NR}}_T \\geq \\mathrm{NR}_T$, making it a strictly stronger metric. We also introduce \\emph{high probability Nash regret} $\\widehat{\\mathrm{NR}}_T = μ^\\star - (\\prod_t μ_{I_t})^{1/T}$, giving the first high probability regret bounds in fair bandits. Our two-phase algorithm, Round Robin Nash Confidence Bound (\\texttt{RR-NCB}), combines round robin exploration with a Nash confidence bound index policy. We show $\\widetilde{\\mathrm{NR}}_T \\leq \\widetilde{\\mathcal{O}}(\\sqrt{k\\log T/T})$ and, with probability $1-δ$, $\\widehat{\\mathrm{NR}}_T \\leq \\widetilde{\\mathcal{O}}(\\sqrt{k\\log(kT/δ)/T})$, matching the optimal $\\widetilde{\\mathcal{O}}(\\sqrt{k/T})$ rate despite the stronger metrics. Optimality follows from a lower bound via AM-GM and standard $k$-armed bandit minimax arguments. Simulations validate our theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07737",
  "author_total": 1,
  "authors": [
    {
      "name": "Avishek Ghosh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8pwh2vj5x81k7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07737",
  "fulltext_url": null,
  "id": "8pwh2vj5x81k7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07737",
  "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": "W7220930971",
  "pdf_url": "https://arxiv.org/pdf/2610.07737",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8pwh2vj5x81k7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:32Z",
      "native_id": "2610.07737",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07737"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Nash Social Welfare for Multi Armed Bandits: Trajectory-wise Expected and High Probability Regret",
  "title_text": "Nash Social Welfare for Multi Armed Bandits: Trajectory-wise Expected and High Probability Regret",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/8pwh2vj5x81k7",
  "venue": "arXiv",
  "versions": []
}