{
  "abstract": "We propose a parameter-free and efficient gradient descent algorithm for $(L_0,L_1)$-smooth optimization. This notion of smoothness, introduced by Zhang et al. (2020), allows the smoothness bound to grow with the gradient norm. Similar to cutting-plane localization, our method uses a sequence of observations to maintain a region of plausible values for $(L_0,L_1)$ and uses this region to select step sizes. In the convex setting, our method attains the standard iteration complexity bound $\\tilde{\\mathcal{O}}\\left(\\left(L_0/ε+ L_1^2\\right)\\|x^0-x^*\\|^2\\right)$ for finding an $ε$-optimal solution, while in the nonconvex setting it finds an $ε$-stationary point within $\\tilde{\\mathcal{O}}\\left(\\left(L_0/ε^2 + L_1/ε\\right)(f(x^0)-f^*)\\right)$ iterations. To the best of our knowledge, this is the first work to establish, for a single parameter-free algorithm, the best-known iteration-complexity guarantee in the nonconvex setting while also providing the above iteration-complexity guarantee in the convex setting. Experiments on logistic regression and real phase retrieval demonstrate that our method can exploit $(L_0,L_1)$-smoothness and perform efficiently on nonconvex problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27681",
  "author_total": 3,
  "authors": [
    {
      "name": "Rikiya Okazaki"
    },
    {
      "name": "Shotaro Yagishita"
    },
    {
      "name": "Hideitsu Hino"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a00gvc3bkr162.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27681",
  "fulltext_url": null,
  "id": "a00gvc3bkr162",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27681",
  "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",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7223147283",
  "pdf_url": "https://arxiv.org/pdf/2610.27681",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a00gvc3bkr162.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27681",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27681"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Adaptive Normalized Gradient Method for $(L_0,L_1)$-smooth Problems: Estimating Problem Parameters as a Region",
  "title_text": "Adaptive Normalized Gradient Method for (L_0,L_1)-smooth Problems: Estimating Problem Parameters as a Region",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/a00gvc3bkr162",
  "venue": "arXiv",
  "versions": []
}