{
  "abstract": "Recent results [arXiv:2602.04378, arXiv:2602.22608] have established worst-case objective-error lower bounds of order $1/T^2$ after $T$ linear minimization oracle (LMO) calls, matching the upper bound [arXiv:1406.1305] in its dependence on $T$ and thereby resolving a long-standing open question. Upon close inspection, however, both lower bounds are somewhat brittle. The first [arXiv:2602.04378] applies only to the standard Frank--Wolfe algorithm, and, as we show, its prescribed slow exact-line-search trajectories can amplify sufficiently small initialization errors exponentially, corresponding to roughly one additional bit of initialization accuracy per iteration. The second [arXiv:2602.22608] obtains lower bounds for any deterministic LMO-based first-order method via a resisting oracle that adversarially adapts to the queries of the algorithm. In both cases it is conceivable that randomization may circumvent these lower bounds; in particular, neither establishes a lower bound for unrestricted randomized first-order LMO methods. We close this gap by proving an $Ω(L\\operatorname{diam}(S)^2/T^2)$ distributional lower bound for randomized first-order linear minimization oracle methods over strongly convex sets. While our construction reuses the family of [arXiv:2602.22608], our proof characterizes the conditional distribution of the hidden permutation after adaptive queries and bounds the expected decrease of a combinatorial statistic controlling optimizer variance. We further show that the same error threshold holds with probability at least $3/4$ and yields a stopping-time tail bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13590",
  "author_total": 1,
  "authors": [
    {
      "name": "Sebastian Pokutta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/983265jssp01g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13590",
  "fulltext_url": null,
  "id": "983265jssp01g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13590",
  "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.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7221521626",
  "pdf_url": "https://arxiv.org/pdf/2610.13590",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/983265jssp01g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13590",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13590"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets",
  "title_text": "Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/983265jssp01g",
  "venue": "arXiv",
  "versions": []
}