{
  "abstract": "Median-of-means (MoM) is a powerful technique that theoretically enables near sub-Gaussian finite-sample rate for parameter estimation when the underlying data distribution is heavy-tailed (e.g., assumed to have only two first finite moments). A recent work has extrapolated this technique to median-of-\\textit{randomized}-U-Statistics (MoRU) and median-of-\\textit{incomplete}-U-Statistics (MoIU) for estimating expectations of heavy-tailed pairwise kernels. In \\citet{pmlr-v97-clemencon19a}, a concentration rate that scales like $O(n^{-1/2})$ with sample size has been proven for MoRU. However, despite the computational advantage of the latter, the analysis of finite-sample bound for MoIU remains a significant theoretical challenge. As noted by the authors, a straightforward application of McDiarmid's inequality yields a loose bound of order $O(n^{-1/4})$. In this work, we prove a finite-sample concentration bound for the MoIU estimator that scales as $O(n^{-1/2})$ with respect to the sample size using a delicate convex decomposition approach. Furthermore, we show that our proof can be seamlessly extended to geometric median in multivariate settings. Using a Serfling-type argument, we extrapolate our results into a regime where data pairs are selected without replacement across blocks, breaking the usual block-wise independence condition. Then, using a Bernstein-type treatment for U-Statistics, we tighten the dependency of our bounds on the margin $τ$ from $O(τ^{-3/2})$ achieved in the previous work to $O(τ^{-1})$. Finally, we proved anti-concentration results that is applicable for all median estimators presented in this work to demonstrate that $M\\le O(n)$ is an intrinsic restriction on block sizes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.00661",
  "author_total": 2,
  "authors": [
    {
      "name": "Nong Minh Hieu"
    },
    {
      "name": "Antoine Ledent"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4kyz0cehkvrk5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.00661",
  "fulltext_url": null,
  "id": "4kyz0cehkvrk5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.00661",
  "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",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7163162632",
  "pdf_url": "https://arxiv.org/pdf/2606.00661",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4kyz0cehkvrk5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:26Z",
      "native_id": "2606.00661",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.00661"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "On Finite-sample Concentration of Median of Incomplete U-Statistics",
  "title_text": "On Finite-sample Concentration of Median of Incomplete U-Statistics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/4kyz0cehkvrk5",
  "venue": "arXiv",
  "versions": []
}