{
  "abstract": "We estimate the conditional population-risk curve of a realized smooth nonconvex gradient flow from the training sample. Flow approximate leave-one-out (Flow-ALO) propagates a deletion response and evaluates omitted observations at approximate deleted paths. The risk-curve error decomposes into response approximation, exact-LOO fluctuation, and deletion-to-full risk transfer. On each fixed finite horizon, bounded centered training-loss gradients, a one-sided Hessian lower bound, locally Lipschitz Hessians, and a strict tube-closure condition yield an explicit $(n-1)^{-2}$ bound for the deletion-response error. Bounded evaluation-loss gradients transfer the deletion-response bound to the score without requiring the Hessian to be invertible. Direct first-order jackknife cancellation and exact-LOO concentration control deletion-to-full risk transfer and fluctuation, respectively, completing recovery of the conditional population-risk curve. For bounded smooth two-layer mean-field networks training both layers, the score-error bound is uniform in width.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30261",
  "author_total": 1,
  "authors": [
    {
      "name": "Mingzhi Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qqd2rymxt8pt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30261",
  "fulltext_url": null,
  "id": "4qqd2rymxt8pt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30261",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7204957427",
  "pdf_url": "https://arxiv.org/pdf/2608.30261",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qqd2rymxt8pt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:54Z",
      "native_id": "2608.30261",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30261"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Estimating Population-Risk Curves Along Nonconvex Gradient Flows from the Training Sample",
  "title_text": "Estimating Population-Risk Curves Along Nonconvex Gradient Flows from the Training Sample",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:54Z",
  "url": "https://research.draconitic.com/paper/4qqd2rymxt8pt",
  "venue": "arXiv",
  "versions": []
}