{
  "abstract": "We study the last iterate of the projected subGradient Method with a constant stepsize for convex $L$-Lipschitz objectives in $\\mathbb{R}^d$, minimized over closed convex sets. Assuming that the initial iterate $x_1$ has distance at most $R$ from a minimizer, we establish matching upper and lower bounds on the minimax optimization error at $x_n$, where the infimum is taken over constant stepsizes. Specifically, this error is of order $L\\cdot R\\cdot\\sqrt{\\min\\{d,\\log n\\}/n}$. Notably, when the dimension is fixed, the minimax error decays as $1/\\sqrt{n}$, without the extra logarithmic factor intrinsic to dimension-free analyses. Our analysis exploits a combinatorial argument based on linear algebra and novel geometric constructions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.15980",
  "author_total": 4,
  "authors": [
    {
      "name": "Guglielmo Beretta"
    },
    {
      "name": "Tommaso Cesari"
    },
    {
      "name": "Roberto Colomboni"
    },
    {
      "name": "Andrea Paudice"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cw2yb5ndprgyt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.15980",
  "fulltext_url": null,
  "id": "cw2yb5ndprgyt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.15980",
  "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": "W7169798721",
  "pdf_url": "https://arxiv.org/pdf/2607.15980",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cw2yb5ndprgyt.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:28Z",
      "native_id": "2607.15980",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.15980"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Minimax Analysis of the Last Iterate of the SubGradient Method in Finite Dimension",
  "title_text": "A Minimax Analysis of the Last Iterate of the SubGradient Method in Finite Dimension",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/cw2yb5ndprgyt",
  "venue": "arXiv",
  "versions": []
}