{
  "abstract": "Modern gradient boosting algorithms are based on Newton's method, which excels empirically but lacks global convergence guarantees. While recent work on Gradient-Regularized Newton (GRN) boosting achieved a global $\\mathcal{O}(1/γ^{12}k^2)$ rate, its analysis was not able to exploit the natural Hessian geometry and suffers from a heavy $γ^{12}$ dependence on a worst-case `weak gradient edge' $γ$. In this paper, we tackle these theoretical drawbacks by generalizing the Affine-Invariant Cubic Newton (AICN) scheme to convex optimization with weak learners. Our analysis is based on average cosine values $\\overlineΘ_k$ along the optimization path and directional improvements $δ_k$ in the Hessian-induced norm. We show that the weak-AICN update reduces to a damped weak-Newton step with an explicit stepsize determined by $δ_k$. Under semi-strong self-concordance, we establish a global $\\mathcal{O}(1/\\overlineΘ_k^4 k^2)$ rate, significantly improving upon the weak-GRN bound. We further prove a local linear rate, global linear convergence for Hessian-dominated losses, cosine angle lower bounds, and present weak-AdaAICN, which adaptively tunes the cubic regularization. Finally, we demonstrate the broader applicability of this framework beyond gradient boosting by showing that Newton coordinate descent emerges naturally as a special case of optimization with weak learners.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33516",
  "author_total": 4,
  "authors": [
    {
      "name": "Nikita Zozoulenko"
    },
    {
      "name": "Daniel Falkowski"
    },
    {
      "name": "Thomas Cass"
    },
    {
      "name": "Lukas Gonon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3qrk0tkde38hy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33516",
  "fulltext_url": null,
  "id": "3qrk0tkde38hy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33516",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7223894492",
  "pdf_url": "https://arxiv.org/pdf/2610.33516",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3qrk0tkde38hy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33516",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33516"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Affine-invariant Cubic Newton with Weak Learners",
  "title_text": "Affine-invariant Cubic Newton with Weak Learners",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:49Z",
  "url": "https://research.draconitic.com/paper/3qrk0tkde38hy",
  "venue": "arXiv",
  "versions": []
}