{
  "abstract": "This paper considers the design of optimal fixed-step first-order methods for high-dimensional minimization of $L$-smooth convex functions. For optimizing worst-case performance measured via suboptimality of the final function value (relative to the initial squared distance to a minimizer), we provide an algebraic proof of the optimality of the optimized gradient method (OGM) and establish its uniqueness among all fixed-step first-order methods. For the alternative measure of final squared gradient norm (relative to initial suboptimality), we prove the OGM-G method is optimal and uniquely so among fixed-step first-order methods. Finally, for the setting measuring the final squared gradient norm (relative to the initial squared distance to a minimizer), we show the recently proposed Lemniscate method is optimal and uniquely so. Our proofs rely on algebraic reductions for lower bound arguments rather than traditional information-theoretic bounds, which were previously only able to establish OGM's optimality but not uniqueness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07671",
  "author_total": 3,
  "authors": [
    {
      "name": "Benjamin Grimmer"
    },
    {
      "name": "Sunghyeon Jo"
    },
    {
      "name": "Chanwoo Park"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/15t5ygerk6ghk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07671",
  "fulltext_url": null,
  "id": "15t5ygerk6ghk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07671",
  "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": "W7212083606",
  "pdf_url": "https://arxiv.org/pdf/2609.07671",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/15t5ygerk6ghk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07671",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07671"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On Minimax Optimality and Uniqueness of Fixed-Step First-Order Methods for Smooth Convex Optimization",
  "title_text": "On Minimax Optimality and Uniqueness of Fixed-Step First-Order Methods for Smooth Convex Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/15t5ygerk6ghk",
  "venue": "arXiv",
  "versions": []
}