{
  "abstract": "We study online vector balancing for independent sparse random vectors: each arrival is supported on a uniformly random set of $s$ of the $n$ coordinates, with nonzero entries equal to one (binary model) or independent random signs (signed model), and an online algorithm must sign each vector upon arrival. The prefix discrepancy is the largest $\\ell_\\infty$-norm of the signed sums up to the horizon $T$, the terminal discrepancy that of the sum at time $T$. For binary inputs with $2\\le s\\le n/2$ and $T=Θ(n)$, Altschuler and Tikhomirov showed that both are of order $\\sqrt{s}+\\log\\log n$; they asked how the prefix discrepancy depends on the horizon and whether a matching terminal bound holds at every prescribed time. Over long horizons, the two differ. For every fixed $s\\ge2$ and $c>0$, every online algorithm, even a randomized one that knows the horizon, incurs with high probability prefix discrepancy $Ω(\\log\\log n+(\\log T/\\log n)^{α_s})$ for $n\\le T\\le e^{cn}$, where $α_s=(2^{s-1}-1)/(2^s-1)$. For $s=2$, quadratic Greedy matches this bound; for fixed $s\\ge3$, we give an online algorithm with prefix discrepancy $O(\\log\\log n+(\\log T/\\log n)^{1/2})$. In contrast, a deterministic algorithm that knows neither the horizon nor the failure probability keeps the terminal discrepancy $O_k(\\sqrt{s}+\\log\\log n)$ at each prescribed time, with failure probability $n^{-k}$ for every fixed $k$, for $2\\le s\\le n/2$ (binary model) and $2\\le s\\le n$ (signed model); for $s=2$, quadratic Greedy itself does so. This answers their terminal question, and the same order is optimal for prefixes at all polynomial horizons $T\\ge n$. The lower bound uses a finite safety game realized within the input law; the upper bounds use moments of elementary symmetric polynomials and, at terminal times, a $\\cosh$ potential or a translated gauge with a Foster inequality at every state.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34410",
  "author_total": 1,
  "authors": [
    {
      "name": "Nicola Wengiel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/da0wwr44djqsx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34410",
  "fulltext_url": null,
  "id": "da0wwr44djqsx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34410",
  "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.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7223909673",
  "pdf_url": "https://arxiv.org/pdf/2610.34410",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/da0wwr44djqsx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34410",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34410"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Online Discrepancy of Sparse Random Vectors over Long Horizons",
  "title_text": "Online Discrepancy of Sparse Random Vectors over Long Horizons",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/da0wwr44djqsx",
  "venue": "arXiv",
  "versions": []
}