{
  "abstract": "We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize stochastic stability through the \\emph{critical learning rate}, defined as the largest learning rate for stable training, and obtain $η_{\\mathrm{SGD}}^{\\mathrm{crit}}\\eqsim 1$, $η_{\\mathrm{Polyak}}^{\\mathrm{crit}}\\eqsim \\min\\{1,B(1-β)\\}$, and $η_{\\mathrm{Nesterov}}^{\\mathrm{crit}}\\eqsim \\min\\{1,B^q(1-β)\\}$, where $B$ is the batch size, $β\\in[0,1)$ is the momentum factor, and \\(q>1\\) is the capacity exponent governing spectral decay. Within the stable region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient through power-law decay to a noise floor. We then optimize the final-step risk under a fixed data budget, obtaining a three-regime phase diagram that reveals how the benefits of momentum change with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the optimal small-batch data-scaling exponent, thereby allowing fewer sequential updates without sacrificing data efficiency. In contrast, Nesterov improves the optimal data-scaling exponent in the large-batch regime through look-ahead-induced suppression of noise amplification. Experiments on PLK regression quantitatively support the theoretical scaling predictions, while neural-network experiments provide qualitative evidence for the predicted batch-size dependence and stability behavior.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02728",
  "author_total": 5,
  "authors": [
    {
      "name": "Jia-Nan Wang"
    },
    {
      "name": "Zixun Huang"
    },
    {
      "name": "Kairui Li"
    },
    {
      "name": "Zilin Wang"
    },
    {
      "name": "Lei Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f1qnx5me1gyc2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02728",
  "fulltext_url": null,
  "id": "f1qnx5me1gyc2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02728",
  "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.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7207576054",
  "pdf_url": "https://arxiv.org/pdf/2609.02728",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f1qnx5me1gyc2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2609.02728",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02728"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Momentum in large-batch training: Polyak enlarges the critical batch size, Nesterov improves data efficiency",
  "title_text": "Momentum in large-batch training: Polyak enlarges the critical batch size, Nesterov improves data efficiency",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:59Z",
  "url": "https://research.draconitic.com/paper/f1qnx5me1gyc2",
  "venue": "arXiv",
  "versions": []
}