{
  "abstract": "We study exact support recovery under $(ε,δ)$-differential privacy in sparse high-dimensional linear regression. We introduce Saturated Propose-test-release, a general mechanism that privately releases the output of a discrete selector with probability one once its stability certificate reaches a finite threshold. Exploiting the coordinatewise geometry of the LASSO, we construct a computable support-stability score. The resulting computationally efficient Saturated LASSO satisfies worst-case $(ε,δ)$-differential privacy and achieves exact support recovery with high probability under explicit regularity and beta-min conditions. Maximizing sparsity-indexed certificates yields an adaptive procedure requiring no sparsity knowledge and having exactly the same finite-sample exact-recovery risk as the oracle fixed-sparsity procedure under common public tuning parameters. We also establish a minimax lower bound explicitly tracking $δ$: under its recovery conditions, Saturated LASSO is minimax optimal up to logarithmic factors in $n$ and $1/δ$ uniformly over $0 < δ\\leq ε/16$; in the broad moderate-$δ$ regime, it further matches the lower-bound $δ$-dependence. A complementary information-theoretic construction with known sparsity attains the lower-bound rates up to constant factors under independent Gaussian design, at exponential computational cost. Simulations and a semi-synthetic study using public American Community Survey covariates illustrate the numerical performance of the proposed procedures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05867",
  "author_total": 2,
  "authors": [
    {
      "name": "Jia Gu"
    },
    {
      "name": "T. Tony Cai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cwtv1fp5d8zqw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05867",
  "fulltext_url": null,
  "id": "cwtv1fp5d8zqw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05867",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Inference"
    ]
  ],
  "openalex": "W7220728866",
  "pdf_url": "https://arxiv.org/pdf/2610.05867",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cwtv1fp5d8zqw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05867",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05867"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Learning Sparse Support under Differential Privacy: Adaptive Algorithms and Minimax Limits",
  "title_text": "Learning Sparse Support under Differential Privacy: Adaptive Algorithms and Minimax Limits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/cwtv1fp5d8zqw",
  "venue": "arXiv",
  "versions": []
}