{
  "abstract": "Understanding generalization of randomized learning algorithms remains a central challenge. Mutual-information and total-variation bounds provide complementary ways to quantify data--hypothesis dependence, but they can behave quite differently in high-dimensional regimes. We study a one-parameter family of dependence measures, the vector-valued $L_p$-norm correlation $V_α$, and derive generalization bounds whose dependence terms interpolate between total variation (TV) at $α=1$ and variance and $χ^2$ related behavior at $α=2$. For $α>2$, the Hölder-conjugate loss moment is below two, yielding heavy-tailed specializations that can remain finite even when the loss variance is infinite. A particularly interpretable specialization occurs at $α=2$, where the bound can be expressed in terms of the variability of the algorithm's output density across datasets, $\\int \\sqrt{\\mathrm{Var}_S[p(w\\mid S)]}~ dw$. We show that the entire $V_α$ family is invariant under smooth bijective reparameterizations of the hypothesis space. We also introduce an adaptive density stability (ADS) condition that implies explicit control of $V_2$, and we verify it for Gibbs-type posteriors under bounded losses (yielding rates that depend on the inverse temperature and sample size). To address scaling, we extend the framework to a conditional formulation using supersamples, analogous in spirit to conditional mutual information (CMI), and discuss evaluations in the output space as a practical surrogate for estimating dependence. Simulations on Bayesian linear regression illustrate how tuning $α$ can change the tightness of the resulting bounds across regimes. Additional supersample experiments on noisy MNIST and on a pretrained ResNet-50 for CIFAR-10 show similar behavior of the projected loss-space bounds beyond the original clean-MNIST setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29425",
  "author_total": 3,
  "authors": [
    {
      "name": "Mohammad Hossein Momeni Hamaneh"
    },
    {
      "name": "Mohammad Mahdi Mojahedian"
    },
    {
      "name": "Mohammad Hossein Yassaee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1d848m27qctz2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29425",
  "fulltext_url": null,
  "id": "1d848m27qctz2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29425",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7223214977",
  "pdf_url": "https://arxiv.org/pdf/2610.29425",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1d848m27qctz2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29425",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29425"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Generalization Bounds via Vector-Valued $L_p$ Correlation $V_α$",
  "title_text": "Generalization Bounds via Vector-Valued L_p Correlation V_α",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/1d848m27qctz2",
  "venue": "arXiv",
  "versions": []
}