{
  "abstract": "Tensor methods offer improved complexity guarantees for nonconvex optimization, but evaluating high-order derivatives and solving the corresponding subproblems can be computationally expensive. In this paper, we develop a proximal point Newton framework that exploits third-derivative smoothness using only gradients and Hessians. Under Hessian Lipschitz continuity, a single-loop proximal point Newton method that alternates between negative-curvature directions and Newton steps finds an $(\\varepsilon_g,\\varepsilon_H)$-approximate second-order stationary point within $\\mathcal{O}(\\max\\{\\varepsilon_g^{-3/2},\\varepsilon_H^{-3}\\})$ second-order oracle evaluations. When the third derivative is also Lipschitz continuous, we introduce an accelerated proximal point Newton method and obtain a dimension-free complexity of $\\widetilde{\\mathcal{O}}(\\max\\{\\varepsilon_g^{-4/3},\\varepsilon_H^{-2}\\})$, matching that of the third-order tensor methods up to logarithmic factors without constructing tensors or solving tensor subproblems. We further develop an adaptive inexact Newton--CG variant that requires no prior knowledge of Lipschitz constants and preserves the improved oracle complexity with high probability. Under uniformly bounded queried Hessians, this variant achieves a Hessian-vector product complexity of $\\widetilde{\\mathcal{O}}(\\max\\{\\varepsilon_g^{-5/3},\\varepsilon_H^{-5/2}\\})$, improving upon the previous $\\widetilde{\\mathcal{O}}(\\max\\{\\varepsilon_g^{-3}\\varepsilon_H^{5/2},\\varepsilon_H^{-7/2}\\})$ result attained by existing second-order methods such as cubic regularized Newton method and Newton--CG method. Preliminary numerical experiments are encouraging.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36256",
  "author_total": 2,
  "authors": [
    {
      "name": "Chang He"
    },
    {
      "name": "Shuzhong Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4q1cea9cgddn1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36256",
  "fulltext_url": null,
  "id": "4q1cea9cgddn1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36256",
  "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",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7224118471",
  "pdf_url": "https://arxiv.org/pdf/2610.36256",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4q1cea9cgddn1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36256",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36256"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Proximal Point Newton Methods for Nonconvex Optimization",
  "title_text": "Proximal Point Newton Methods for Nonconvex Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/4q1cea9cgddn1",
  "venue": "arXiv",
  "versions": []
}