{
  "abstract": "Alternating gradient descent-ascent (AltGDA) is a simple and practically effective method for solving finite two-player zero-sum matrix games. However, the theory of AltGDA remains limited: existing results either apply only to unconstrained settings or require restrictive assumptions on the equilibrium in constrained settings. We show that AltGDA converges globally at an $O(1/T)$ ergodic rate in every finite two-player zero-sum matrix game. Unlike prior results, our guarantee holds for every initialization and every horizon $T$: the uniform averages of the AltGDA iterates satisfy an $O(1/T)$ duality-gap bound. Our proof is inspired by numerical results obtained using a novel performance estimation programming (PEP) framework for Lyapunov function search over compact convex sets. Additionally, we provide simple counterexamples showing that the last-iterate duality gap of AltGDA does not converge to zero. This justifies why averaging of iterates is indeed necessary to achieve an $O(1/T)$ rate. We have formalized and machine-checked our global ergodic convergence result in Lean 4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32995",
  "author_total": 4,
  "authors": [
    {
      "name": "Tianlong Nan"
    },
    {
      "name": "Garud Iyengar"
    },
    {
      "name": "Christian Kroer"
    },
    {
      "name": "Shuvomoy Das Gupta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7bh92ep87dqxd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32995",
  "fulltext_url": null,
  "id": "7bh92ep87dqxd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32995",
  "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",
      "cs.GT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214803746",
  "pdf_url": "https://arxiv.org/pdf/2609.32995",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7bh92ep87dqxd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32995",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32995"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "AltGDA Achieves Global $O(1/T)$ Ergodic Convergence in Matrix Games",
  "title_text": "AltGDA Achieves Global O(1/T) Ergodic Convergence in Matrix Games",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/7bh92ep87dqxd",
  "venue": "arXiv",
  "versions": []
}