{
  "abstract": "Recent observations that deep learning models train at the edge of stability have renewed interest in understanding stability properties of classical optimization algorithms. For quadratic objectives, standard gradient descent has a sharp stability threshold: it converges from every initialization when $η<2/L$, diverges from almost every initialization when $η>2/L$, and diverges from every nonzero initialization when $η>2/μ$. For non-Euclidean gradient descent, the corresponding known guarantees are weaker: convergence holds for every initialization when $η<2/S$, where $S$ is the generalized sharpness, while every nonzero initialization diverges when $η>2/μ$. The behavior in the intermediate regime $2/S<η<2/μ$ is not fully characterized. We show that Hilbertian norms are the only ones for which $2/S$ is an almost-everywhere sharp threshold for every positive-definite quadratic objective. Using linear conjugacy between non-Euclidean and Euclidean gradient descent, we show that for general norms the intermediate regime can contain arbitrarily many edges of stability associated with distinct attracting directions. We identify a forward-invariant region of exponentially divergent directions beyond the generalized stability threshold. When the directions of minimum curvature are finite and nondegenerate, we prove that, sufficiently close to $2/μ$ from below, only their invariant directions remain convergent. We give smooth two-dimensional constructions with arbitrarily many distinct thresholds, arbitrarily large relative threshold separation and show these properties to be stable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15223",
  "author_total": 2,
  "authors": [
    {
      "name": "Peter Zaika"
    },
    {
      "name": "Zakhar Shumaylov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/77x08tbae6sjv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15223",
  "fulltext_url": null,
  "id": "77x08tbae6sjv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15223",
  "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"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7221830397",
  "pdf_url": "https://arxiv.org/pdf/2610.15223",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/77x08tbae6sjv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15223",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15223"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Non-Euclidean Gradient Descent Has Many Edges of Stability",
  "title_text": "Non-Euclidean Gradient Descent Has Many Edges of Stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/77x08tbae6sjv",
  "venue": "arXiv",
  "versions": []
}