{
  "abstract": "We consider unconstrained minimization of a possibly nonconvex function $f:\\mathbb{R}^n\\to\\mathbb{R}$ whose $p$th derivative is Lipschitz continuous, with $p\\geq3$, when derivative information is available only through order $p-2$. The unavailable tensors $\\nabla^{p-1}f$ and $\\nabla^pf$ are reconstructed from central finite differences of $\\nabla^{p-2}f$. Since the two reconstructions have different point-query costs, we use a block length $m$ for the order-$p$ approximation and a subblock length $\\ell\\leq m$ for the order-$(p-1)$ approximation. The latter is transported between consecutive reconstruction points using the stored order-$p$ approximation. This leads to an adaptive regularization method that does not require prior knowledge of the Lipschitz constant. We prove finite termination at an $ε$-approximate first-order stationary point and derive an explicit bound on the number of bundled order-$(p-2)$ oracle queries. For $m=(p-2)n^2$ and $\\ell=(p-2)n$, the accuracy-dependent term has dimension--accuracy scaling $n^{3/p}ε^{-(p+1)/p}$. Thus the method retains the $ε^{-(p+1)/p}$ dependence of full order-$p$ regularization while using derivatives only through order $p-2$. Within the dense reconstruction framework considered here, using separate block and subblock lengths gives dimension dependence of order $n^{3/p}$, compared with $n^{4/p}$ when a common length is imposed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01819",
  "author_total": 1,
  "authors": [
    {
      "name": "Alireza Kabgani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/94s6mnyc0s3zr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01819",
  "fulltext_url": null,
  "id": "94s6mnyc0s3zr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01819",
  "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",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01819",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/94s6mnyc0s3zr.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01819",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Multi-rate high-order regularization with an order-$(p-2)$ oracle",
  "title_text": "Multi-rate high-order regularization with an order-(p−2) oracle",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/94s6mnyc0s3zr",
  "venue": "arXiv",
  "versions": []
}