{
  "abstract": "We study online discrepancy minimization for independent random vectors in $\\mathbb{R}^n$ with independent coordinates. Our deterministic algorithms require no knowledge of the horizon, and their guarantees hold at every prescribed deterministic terminal time $T$. For unit-variance sub-exponential coordinates with $ψ_1$-norms at most $σ$, we obtain terminal discrepancy $O(σ^4\\sqrt n)$ with failure probability $\\exp(-Ω(σ^3\\sqrt n))$. Independently masking centered unit-variance marks from the same class with retention probability $k/n$ gives $O(σ^4\\sqrt k)$ discrepancy with failure probability $\\exp(-Ω(σ^3\\sqrt k))$ when $k\\gtrsim\\log^2(2n)$. The fourth-power dependence on $σ$ is necessary for these sub-exponential classes at the stated confidence levels. For standard Gaussian marks, we obtain $O(\\sqrt k+\\sqrt{\\log(2n)})$ discrepancy with failure probability $\\exp(-Ω(\\sqrt k+\\sqrt{\\log(2n)}))$. Both terms are necessary up to absolute constants for $T\\ge n$. In particular, $k\\gtrsim\\log n$ is the optimal sparsity threshold for $O(\\sqrt k)$ discrepancy. For centered unit-variance sub-Gaussian marks with any fixed $ψ_2$-norm bound $σ$, logarithmic sparsity suffices for $O(σ\\sqrt{k\\log(e+σ)})$ discrepancy with probability tending to one as $n\\to\\infty$. For a dimension-dependent family of unit-variance laws with rare activations, matching bounds determine the fixed-confidence online discrepancy order $Θ(\\sqrt n\\log\\log n)$ at sufficiently long prescribed horizons. Our dense result generalizes the Rademacher-input theorem of Bansal and Spencer (2020) and establishes the horizon-independent $O(\\sqrt n)$ Gaussian terminal discrepancy scale anticipated by Gamarnik et al. (2022). The proofs combine regularization, global Fenchel estimates, coupled Lyapunov functions, and terminal moment recursions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.10040",
  "author_total": 1,
  "authors": [
    {
      "name": "Nicola Wengiel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7mfrhvv9f92d6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.10040",
  "fulltext_url": null,
  "id": "7mfrhvv9f92d6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.10040",
  "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": "W7202252298",
  "pdf_url": "https://arxiv.org/pdf/2608.10040",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7mfrhvv9f92d6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2608.10040",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.10040"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Online Discrepancy for Sub-Exponential Product Distributions via Regularization",
  "title_text": "Online Discrepancy for Sub-Exponential Product Distributions via Regularization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7mfrhvv9f92d6",
  "venue": "arXiv",
  "versions": []
}