{
  "abstract": "How much data must a fixed learner retain? Hanneke, Moran, Shlimovich and Yehudayoff (COLT 2025) posed this question for linear regression with the minimum-norm empirical risk minimizer. A selector sees a finite dataset $D\\subseteq R^d\\times R$, keeps at most $n$ examples with nonnegative weights, and $F_w(d,n)$ is the worst-case ratio between the full-data loss of the trained predictor and the optimal loss. The value is $\\infty$ for $n<d$, $d+1$ at $n=d$ and $1$ for $n\\ge2d$, and the regime $d<n<2d$ was left open. We settle several cases. For every $d$ we prove $F_w(d,2d-1)=1+1/d$, which confirms a claim stated without proof in the original note. We also prove $F_w(3,4)=5/3$, $F_w(4,5)=2$ and $F_w(4,6)=3/2$, the three smallest cells not covered by that formula. For every intermediate budget $n=d+k$ we prove the lower bound $F_w(d,d+k)\\ge1+Γ_{d,k}$, where $Γ_{d,k}$ is an explicit harmonic quantity over balanced partitions of $d$. This bound is the exact minimax value on the class of datasets whose whitened systems split into orthogonal circuit blocks. All proved values equal $1+Γ_{d,k}$, and we conjecture that this holds throughout the open regime. Our upper bounds combine a rigidity theorem for positive spanning configurations of loss gradients with normal forms of the small positive bases in $R^3$ and $R^4$. These forms are special cases of the classification of Cornaz, Kerleau and Royer; we also control all gradients outside the basis. A dimension-free extremal-basis argument converts sign-cone geometry into selections of $d+1$ points. Explicit counterexamples rule out several shorter routes. Every upper bound is constructive, with selection procedures polynomial in the number of points for fixed dimension. Every numerical claim about a specific instance is an exact rational or algebraic identity, recomputed in exact arithmetic in the supplementary material.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28007",
  "author_total": 1,
  "authors": [
    {
      "name": "Guangjian Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qpfp2z355w54.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28007",
  "fulltext_url": null,
  "id": "4qpfp2z355w54",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28007",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7204761484",
  "pdf_url": "https://arxiv.org/pdf/2608.28007",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qpfp2z355w54.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2608.28007",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28007"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Exact Risk Ratios for Weighted Data Selection in Linear Regression",
  "title_text": "Exact Risk Ratios for Weighted Data Selection in Linear Regression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:56Z",
  "url": "https://research.draconitic.com/paper/4qpfp2z355w54",
  "venue": "arXiv",
  "versions": []
}