{
  "abstract": "We study distributed one-dimensional mean estimation under a 1-bit communication constraint. Each agent observes one sample, drawn independently from an unknown distribution, and returns a single bit in response to a query $Q: \\mathbb{R}\\to\\{0,1\\}$ chosen by a central learner. The distribution has mean in $[-λ,λ]$ and $k$-th central moment at most $σ^k$, for a fixed $k>1$. The order-optimal two-stage protocol of Lau and Scarlett uses responses from the first batch to choose the second-batch queries, motivating the question of whether this single round of interaction is necessary. We answer this negatively: for every $k>1$, a non-adaptive protocol attains the adaptive 1-bit minimax rate (and concurrent works reached the same conclusion via different strategies). We further determine the minimax sample complexity among non-adaptive 1-bit estimators when every one-set $Q^{-1}(1)$ is restricted to a union of at most $s$ intervals. Relative to unrestricted non-adaptive 1-bit querying, this constraint adds a term of order $(λσ/(s\\varepsilon^2))\\log(1/δ)$, giving the full tradeoff between sample complexity and interval complexity to within $k$-dependent constant factors. As a corollary, we identify, order-wise, the minimum interval budget needed to retain the unrestricted 1-bit minimax sample rate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08564",
  "author_total": 2,
  "authors": [
    {
      "name": "Ivan Lau"
    },
    {
      "name": "Jonathan Scarlett"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dkwmnr481dtxb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08564",
  "fulltext_url": null,
  "id": "dkwmnr481dtxb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08564",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Privacy-Preserving Technologies in Data"
    ]
  ],
  "openalex": "W7211989088",
  "pdf_url": "https://arxiv.org/pdf/2609.08564",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dkwmnr481dtxb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08564",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08564"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Non-Adaptive 1-Bit Mean Estimation: Minimax Rates and the Sample-Interval Tradeoff",
  "title_text": "Non-Adaptive 1-Bit Mean Estimation: Minimax Rates and the Sample-Interval Tradeoff",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/dkwmnr481dtxb",
  "venue": "arXiv",
  "versions": []
}