{
  "abstract": "Existing theory of momentum assumes that gradients arrive at every parameter at a roughly constant rate, an assumption violated in practice by heavy-tailed data distributions and modern architectures. We theoretically analyze the dynamics of two tractable models of momentum under sparse updates: a least squares model with sparse inputs and a logistic regression model with a rare class. Both admit exact closed-form second-moment dynamics whose high-dimensional limits we characterize across three scaling exponents for sparsity, batch size, and momentum decay. The phase structure on both problems is governed by the ratio of two intrinsic timescales: a momentum retention timescale (how many active updates the buffer survives) and a learning timescale (how many active updates it takes to reduce the squared error). When learning is much slower than retention, the limit matches SGD; when learning is faster, the system is unstable; where the timescales coincide, we recover classical heavy-ball dynamics. The oscillatory dynamics occur at different momentum values for different token sparsity, creating a spectral conflict for global momentum across token frequencies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.28961",
  "author_total": 2,
  "authors": [
    {
      "name": "Katie Everett"
    },
    {
      "name": "Elliot Paquette"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7pb4cpjy4em2y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.28961",
  "fulltext_url": null,
  "id": "7pb4cpjy4em2y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.28961",
  "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",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7162773681",
  "pdf_url": "https://arxiv.org/pdf/2605.28961",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7pb4cpjy4em2y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2605.28961",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.28961"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Dynamics of Stochastic Momentum with Sparse Updates in High Dimensions",
  "title_text": "Dynamics of Stochastic Momentum with Sparse Updates in High Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:38Z",
  "url": "https://research.draconitic.com/paper/7pb4cpjy4em2y",
  "venue": "arXiv",
  "versions": []
}