{
  "abstract": "The classical perceptron cycling theorem of Block and Levin \\cite{BlockLevin1970} bounds correction sequences whose selected updates come from a finite set and have nonpositive inner product with the current state. We prove a robust variant for additive trajectories $z_{k+1}=z_k+u_k$, for all integers $k\\geq0$, with increments in a finite set $U\\subset E$, where $E$ is a finite-dimensional real inner-product space and $0\\in\\conv U$. Let $A\\colon E\\to E$ have positive-definite symmetric part, without requiring symmetry, and let $B\\geq0$. If each $u_k$ is a $B$-approximate minimizer of $u\\mapsto\\ip{Az_k}{u}$ over $U$, then $\\sup_{k\\geq0}\\norm{z_k}\\leq C(1+\\norm{z_0}+B)$, with $C=C(E,U,A)$ independent of the initial state, $B$, and all admissible update choices. We derive two algorithmic consequences. The first is an $O(k^{-1})$ last-iterate norm bound for harmonic vertex-returning Frank--Wolfe for affine strongly monotone variational inequalities on polytopes with relatively interior solutions. It extends the quadratic Frank--Wolfe/herding guarantee of Bach, Lacoste-Julien, and Obozinski \\cite[Section~4.2]{BachEtAl2012} to nonsymmetric affine operators. The second consequence concerns oblique relaxation for linear inequalities. Greedy corrections through a fixed matrix with positive-definite symmetric part remain bounded even for inconsistent systems. In particular, for a square matrix $G$ with positive-definite symmetric part, repeatedly increasing the coordinate corresponding to a most-violated inequality of $Gx\\geq b$ terminates at an exactly feasible point after finitely many unit corrections. This remains true under bounded additive selection errors, provided the stopping test uses the true inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18342",
  "author_total": 1,
  "authors": [
    {
      "name": "Tobias Harks"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9wax1qj285gx6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18342",
  "fulltext_url": null,
  "id": "9wax1qj285gx6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18342",
  "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",
      "cs.GT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7213529570",
  "pdf_url": "https://arxiv.org/pdf/2609.18342",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9wax1qj285gx6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.18342",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18342"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Robust Perceptron Cycling Theorem and Applications",
  "title_text": "A Robust Perceptron Cycling Theorem and Applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/9wax1qj285gx6",
  "venue": "arXiv",
  "versions": []
}