{
  "abstract": "Cyclic coordinate descent is widely used to train support vector machines, but global linear convergence alone gives limited guidance about the rate on a particular instance. We study exact cyclic coordinate minimization for box-constrained convex quadratics and develop global contraction bounds that capture both coordinate order and boundary geometry. Our convergence certificate combines two ingredients: interactions between successive coordinate updates, encoded by the triangular part of the normalized Hessian, and objective growth, captured by a matrix lower bound on the optimality gap. It remains valid for cases with singular Hessians, nonunique minimizers, and updates clipped at the boundary. An error-bound specialization of our analysis strictly sharpens the Wang-Lin factor when both use the same error-bound constant. In the positive-definite case, the analysis shows that the worst-case contraction of one Gauss-Seidel sweep remains a global upper bound under box constraints, with equality for an interior optimizer. We also demonstrate the limitations of Hessian-only bounds by constructing a two-coordinate family with a fixed singular normalized Hessian, whose worst-case contraction approaches one as its boundary margin vanishes. The theory is complemented by practical procedures for bounding the worst-case contraction on individual instances and numerical evaluation on several classical SVM datasets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39240",
  "author_total": 2,
  "authors": [
    {
      "name": "Egor Gladin"
    },
    {
      "name": "Eva Ibodova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ah9m1mp8zrwdq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39240",
  "fulltext_url": null,
  "id": "ah9m1mp8zrwdq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39240",
  "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": "W7215101571",
  "pdf_url": "https://arxiv.org/pdf/2609.39240",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ah9m1mp8zrwdq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39240",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39240"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Boundary Geometry and Global Rates of Cyclic Coordinate Descent for SVM Duals",
  "title_text": "Boundary Geometry and Global Rates of Cyclic Coordinate Descent for SVM Duals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/ah9m1mp8zrwdq",
  "venue": "arXiv",
  "versions": []
}