{
  "abstract": "We propose and analyze several inexact regularized Newton-type methods for finding a global saddle point of \\emph{convex-concave} unconstrained min-max optimization problems. Compared to first-order methods, our understanding of second-order methods for min-max optimization is relatively limited, as obtaining global rates of convergence with second-order information is much more involved. In this paper, we examine how second-order information can be used to speed up extra-gradient methods, even under inexactness. Specifically, we show that the proposed methods generate iterates that remain within a bounded set and that the averaged iterates converge to an $\\epsilon$-saddle point within $O(\\epsilon^{−2/3})$ iterations in terms of a restricted gap function. This matched the theoretically established lower bound in this context. We also provide a simple routine for solving the subproblem at each iteration, requiring a single Schur decomposition and $O(\\log\\log(1/\\epsilon))$ calls to a linear system solver in a quasi-upper-triangular system. Thus, our method improves the existing line-search-based second-order min-max optimization methods by shaving off an $O(\\log\\log(1/ϵ))$ factor in the required number of Schur decompositions. Finally, we present numerical experiments on synthetic and real data that demonstrate the efficiency of the proposed methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2210.12860",
  "author_total": 3,
  "authors": [
    {
      "name": "Tianyi Lin"
    },
    {
      "name": "Panayotis Mertikopoulos"
    },
    {
      "name": "Michael I. Jordan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5b1gtnxzx3j14.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "5b1gtnxzx3j14",
  "issns": [
    "2835-8856"
  ],
  "landing_url": "https://hal.science/hal-03873989v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-oc"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-03873989/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-05-26",
  "publisher": "[Amherst Massachusetts]: OpenReview.net, 2022",
  "ris": "https://research.draconitic.com/paper/5b1gtnxzx3j14.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:59Z",
      "native_id": "hal-03873989",
      "source": "hal",
      "url": "https://hal.science/hal-03873989v2"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-oc",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Explicit second-order min-max optimization: Practical algorithms and complexity analysis",
  "title_text": "Explicit second-order min-max optimization: Practical algorithms and complexity analysis",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:17:59Z",
  "url": "https://research.draconitic.com/paper/5b1gtnxzx3j14",
  "venue": "Transactions on Machine Learning Research",
  "versions": []
}