{
  "abstract": "Constant-step stochastic gradient descent with momentum (SGDM) augments the parameter recursion with a momentum state that carries geometrically decaying memory of past stochastic gradients. We study this coupled recursion as a nonlinear time series and develop a framework for its stationary behavior and distributional limits. The joint parameter-momentum process forgets its initialization geometrically under a finite $p$th moment for $p > 1$, yielding a unique stationary causal solution. Under a second-moment condition, this contraction leads to an invariance principle and quenched Gaussian limit for averaged SGDM, while a small-learning-rate analysis gives a Gaussian limit for the last iterate. In a stochastic-curvature quadratic model, we further study how the rate of initialization forgetting relates to stationary variance and averaged fluctuations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16034",
  "author_total": 3,
  "authors": [
    {
      "name": "Younghun Lee"
    },
    {
      "name": "Jake Zummo"
    },
    {
      "name": "Wei Biao Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3e1h4vyy5y1ap.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16034",
  "fulltext_url": null,
  "id": "3e1h4vyy5y1ap",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16034",
  "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": "W7221802339",
  "pdf_url": "https://arxiv.org/pdf/2610.16034",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3e1h4vyy5y1ap.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:26Z",
      "native_id": "2610.16034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A Nonlinear Time-Series Theory for Constant-step SGD with Momentum",
  "title_text": "A Nonlinear Time-Series Theory for Constant-step SGD with Momentum",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/3e1h4vyy5y1ap",
  "venue": "arXiv",
  "versions": []
}