{
  "abstract": "For convex $(L_0,L_1)$-smooth functions, the curvature may grow together with the gradient norm. Therefore, even when $\\|x^0-x^*\\|\\le R$, the gradient and the optimality gap at the starting point may be exponentially large in $L_1R$, and the bound $\\sqrt{L_0R^2/\\varepsilon}$, known for the accelerated method on functions with Lipschitz gradient, no longer describes the complexity of the problem. We determine the oracle complexity of this class for arbitrary deterministic first-order methods: for $L_0R^2/\\varepsilon\\ge1$, the optimal number of oracle calls $N$ satisfies $1+N=\\widetildeΘ(\\sqrt{L_0R^2/\\varepsilon}+L_1R)$. The upper bound reads $O(\\sqrt{L_0R^2/\\varepsilon}+(1+L_1R)\\log(2+L_1R))$: the method first passes through the large-gradient regime in $O(1+L_1R)$ iterations with an inexact line search, and then the standard accelerated method with a constant stepsize takes over. The lower bound is obtained by means of chains and the orthogonal invariance principle on functions conjugate to strongly convex functions on a polytope. If the initial optimality gap is at most $4\\varepsilon$, the optimal bound becomes $1+N=\\widetildeΘ(\\sqrt{L_0R^2/\\varepsilon}+(L_1R)^{2/3})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21461",
  "author_total": 4,
  "authors": [
    {
      "name": "Mikhail Rudomanov"
    },
    {
      "name": "Ivan Fomin"
    },
    {
      "name": "Alexander V. Gasnikov"
    },
    {
      "name": "Fedor S. Stonyakin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dkhbps7ar1kzz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21461",
  "fulltext_url": null,
  "id": "dkhbps7ar1kzz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21461",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7222509969",
  "pdf_url": "https://arxiv.org/pdf/2610.21461",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dkhbps7ar1kzz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21461",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21461"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Descending Everest with First-Order Oracles: Oracle Complexity of Convex Optimization under $(L_0,L_1)$-Smoothness",
  "title_text": "Descending Everest with First-Order Oracles: Oracle Complexity of Convex Optimization under (L_0,L_1)-Smoothness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/dkhbps7ar1kzz",
  "venue": "arXiv",
  "versions": []
}