{
  "abstract": "We develop a novel generalization of McDiarmid's concentration inequality and use it to obtain a uniform law of large numbers result and error bounds on stochastic optimization problems which simultaneously incorporate two key features: 1) They capture the benefit of training sample reuse in algorithms that also employ easily-sampled auxiliary random variables. 2) They allow for one-component mean-difference bounding functions which are both unbounded in the component of interest and have non-uniform dependence on another component. We demonstrate the utility of our method by deriving concentration inequalities for score-based generative modeling. This application naturally leverages both key features of our new tools: 1) It utilizes auxiliary Gaussian random variables in conjunction with the training samples; accounting for training sample reuse results in tighter error bounds by effectively integrating out the dependence on the auxiliary random variables. 2) The ability to handle non-uniform mean-difference bounding functions allows us to treat unbounded score models and data distributions with unbounded support.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17159",
  "author_total": 1,
  "authors": [
    {
      "name": "Jeremiah Birrell"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2mcrcp2zyr6j4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17159",
  "fulltext_url": null,
  "id": "2mcrcp2zyr6j4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17159",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7221802355",
  "pdf_url": "https://arxiv.org/pdf/2610.17159",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2mcrcp2zyr6j4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:26Z",
      "native_id": "2610.17159",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17159"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A Generalized McDiarmid Inequality: Sample Reuse and Non-Uniform Unbounded Differences",
  "title_text": "A Generalized McDiarmid Inequality: Sample Reuse and Non-Uniform Unbounded Differences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/2mcrcp2zyr6j4",
  "venue": "arXiv",
  "versions": []
}