{
  "abstract": "We study mistake bounds for differentially private online learning and online prediction under oblivious realisable adversaries. Online learning requires the learner to release a hypothesis at each time step whereas in online prediction, the learner only needs to make predictions without releasing a hypothesis. Using a novel lower bound for private online learning and an upper bound for private prediction, we show that the sample complexity of these two problems are separated by a factor that grows with the time horizon for every class of finite Littlestone dimension $d$. First, we prove that every $(ε,δ)$-private online learner has a deterministic realisable stream of length $T$ on which the mistake bound is at least $\\mathbb E [M_T]= O\\left(\\frac dε\\log^{2/3} T\\right)$. In particular, this is the first non-trivial lower in the range $1/T<δ<1/\\log T)$ left open in earlier works[SR22,DSS24,LWY24]. Second, we prove that for every class of of Littlestone dimension $d$, there exists an $(ε,δ)$-jointly private predictor with at most $2^{2^{cd^2}}ε^{-2}\\log^2(2/(εδ))$ expected mistakes, independently of $T$, for some absolute constant $c>0$. Thus, for every fixed class of finite Littlestone dimension when $δ=Θ(1/\\log T)$, private learning requires $O\\left(\\log^{2/3} T\\right)$ expected mistakes, whereas private prediction admits $O(\\log\\log T^2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06822",
  "author_total": 1,
  "authors": [
    {
      "name": "Amartya Sanyal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2mxd3ag3b29xq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06822",
  "fulltext_url": null,
  "id": "2mxd3ag3b29xq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06822",
  "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.CR"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Privacy-Preserving Technologies in Data"
    ]
  ],
  "openalex": "W7220654434",
  "pdf_url": "https://arxiv.org/pdf/2610.06822",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2mxd3ag3b29xq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:32Z",
      "native_id": "2610.06822",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06822"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Private online learning and prediction for Littlestone classes",
  "title_text": "Private online learning and prediction for Littlestone classes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/2mxd3ag3b29xq",
  "venue": "arXiv",
  "versions": []
}