{
  "abstract": "The central flow of Cohen et al. (2025) is an empirically accurate continuous-time model of gradient descent at the edge of stability in deep learning, However, its derivation is heuristic. We propose a perturbative regime in which the central flow is the limit of gradient descent: we assume that the loss decomposes as f = g + εh; in the limit ε → 0, the dynamics of gradient descent with learning rate η converge to the gradient flow of h constrained to the minimizers of g of sharpness at most 2/η. Our approach is formal rather than rigorous; it treats gradient descent as a singularly perturbed dynamical system in ε. Three timescales emerge: a fast timescale of oscillations along the sharpest direction, an intermediate timescale of the self-stabilization mechanism, and a slow timescale of the dynamics along the minimizers of g-the central flow. Using the method of multiple scales, a classical formal method from singular perturbation theory, we derive the expansion of the dynamics in ε: the central flow emerges as the leading-order term in the expansion, while the self-stabilization mechanism appears in the next-order term. We study this mechanism beyond previous analyses: with a single eigenvalue at the edge of stability, we compute the slow drift of the energy of the fluctuations; with several eigenvalues at the edge of stability, we derive the self-stabilization system and explain why fluctuations persist.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01034",
  "author_total": 1,
  "authors": [
    {
      "name": "Raphaël Berthier"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84x4qnwmkdzra.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01034",
  "fulltext_url": null,
  "id": "84x4qnwmkdzra",
  "issns": [],
  "landing_url": "https://hal.science/hal-05733956v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
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      "arxiv",
      "math.OC"
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      "stat.ML"
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      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7206188473",
  "pdf_url": "https://hal.science/hal-05733956/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84x4qnwmkdzra.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:24Z",
      "native_id": "hal-05733956",
      "source": "hal",
      "url": "https://hal.science/hal-05733956v1"
    },
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "The Multiple Timescales of Gradient Descent on the Edge of Stability: A Perturbative Derivation of the Central Flow",
  "title_text": "The Multiple Timescales of Gradient Descent on the Edge of Stability: A Perturbative Derivation of the Central Flow",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:10Z",
  "url": "https://research.draconitic.com/paper/84x4qnwmkdzra",
  "venue": "HAL",
  "versions": []
}