{
  "abstract": "For linear regression with isotropic Gaussian features, n samples, p parameters, signal energy r² and noise variance σ², I prove the exact finite-sample excess risk of the least-squares or minimum-norm interpolating estimator. It is σ²p/(n−p−1) for p ≤ n−2 and r²(1−n/p) + σ²n/(p−n−1) for p ≥ n+2. The proofs use the mean of the inverse Wishart distribution and the rotational invariance of random projections. The formulas reproduce the Hastie–Montanari–Rosset–Tibshirani asymptotics as n, p → ∞ and explain the risk spike at p ≈ n. Simulations with n = 100 and r²/σ² = 4 (400 replications per point) agree with the formulas to within two standard errors wherever |p − n| ≥ 5. Near p = n they become unstable, and I show this is predicted: the second moment of the risk is infinite when |p − n| ≤ 3. At p/n = 2 the min-norm risk is 0.75. That is below the null risk r² = 1 but about 17% above ridge regression at the Bayes-optimal penalty (0.65). In this model, \"benign\" overparameterisation is a statement about the min-norm estimator, not about the value of interpolation.",
  "abstract_shown": true,
  "affiliations": [
    "Former Officer, Chaitanya Godavari Grameena Bank (CGGB)"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Manoj Kumar Aguru"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99m8vqfwzzjs7.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23450083",
  "fulltext_url": null,
  "id": "99m8vqfwzzjs7",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23450083",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23450083/files/10_ai_double_descent_exact.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99m8vqfwzzjs7.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:21:13Z",
      "native_id": "23450083",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23450083"
    }
  ],
  "subjects": [],
  "title": "Exact Finite-Sample Risk of Minimum-Norm Interpolation: Proofs of the Double-Descent Curve for Isotropic Gaussian Least Squares, with Simulation Checks and the Ridge Comparison",
  "title_text": "Exact Finite-Sample Risk of Minimum-Norm Interpolation: Proofs of the Double-Descent Curve for Isotropic Gaussian Least Squares, with Simulation Checks and the Ridge Comparison",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:13Z",
  "url": "https://research.draconitic.com/paper/99m8vqfwzzjs7",
  "venue": "Zenodo",
  "versions": []
}