{
  "abstract": "Many learning problems couple a high-dimensional block of parameters with a handful of adversarial or dual variables: worst-group risk over a few groups, or learning under a few constraints. We consider $\\min_{x\\in X}\\max_{y\\in Y} f(x,y)$ with $X\\subseteq\\mathbb{R}^n$, a compact convex set and strongly convex with condition number $κ$, and we count the oracle calls for $x$ and for $y$ separately. The textbook approach runs a cutting-plane method in $y$ and solves every inner problem to accuracy $\\varepsilon$ with an accelerated method; it needs $O(m\\log(1/\\varepsilon))$ calls for $y$ but $O(m\\sqrtκ\\log^2(1/\\varepsilon))$ calls for $x$. We show that the inner problems need not be solved accurately. Our method, certificate transport, keeps a strongly convex lower model of a single slice and uses it as a prior for a short accelerated run on the next slice. By concavity in $y$, every call for $y$ then either cuts the localizer or moves the lower model to a mixture of the two slices with a certified increase of the lower bound. For $m=1$ this gives an $\\varepsilon$-saddle point after $O(\\sqrtκ\\log(1/\\varepsilon))$ calls for $x$, up to an initialization term, and $O(\\log(1/\\varepsilon))$ calls for $y$; both counts are optimal, even though $f(x,\\cdot)$ is only assumed to be concave and Lipschitz. For general $m$, with centers of gravity of the localizer treated as computable, the method needs $O((m+\\sqrt{mκ})\\log(1/\\varepsilon))$ calls for $x$ and the optimal $O(m\\log(1/\\varepsilon))$ calls for $y$. Under strong concavity in $y$, a two-point accelerated method makes the number of calls for $x$ independent of $m$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32441",
  "author_total": 2,
  "authors": [
    {
      "name": "Ivan Fomin"
    },
    {
      "name": "Alexander V. Gasnikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bkxwb0n7nrwnn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32441",
  "fulltext_url": null,
  "id": "bkxwb0n7nrwnn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32441",
  "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": "W7214768246",
  "pdf_url": "https://arxiv.org/pdf/2609.32441",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bkxwb0n7nrwnn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32441",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32441"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Saddle-Point Problems with a Low-Dimensional Block Do Not Need Accurate Inner Solves",
  "title_text": "Saddle-Point Problems with a Low-Dimensional Block Do Not Need Accurate Inner Solves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/bkxwb0n7nrwnn",
  "venue": "arXiv",
  "versions": []
}