{
  "schema_version": 1,
  "title": "Reproduction: Non-Euclidean Gradient Descent Operates at the Edge of Stability",
  "emoji": "📊",
  "space_id": "snaykey/repro-noneuclidean-gd-eos",
  "paper": {
    "openreview_id": "piWlEHb4Db"
  },
  "tags": [
    "icml2026-repro",
    "paper-piWlEHb4Db"
  ],
  "updated_at": "2026-08-01T00:00:00Z",
  "root": {
    "slug": "index",
    "title": "Reproduction: Non-Euclidean Gradient Descent Operates at the Edge of Stability",
    "file": "pages/index.md",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "file": "pages/executive-summary/page.md",
        "children": []
      },
      {
        "slug": "claim-1-def21-directional-smoothness-iff",
        "title": "Directional smoothness D is defined so that, for non-Euclidean gradient descent with step size eta, the loss decreases if and only if D <= 2/eta (Definition 2.1).",
        "file": "pages/claim-1-def21-directional-smoothness-iff/page.md",
        "children": []
      },
      {
        "slug": "claim-2-def22-generalized-sharpness",
        "title": "Generalized sharpness S is defined as the maximum of d^T times the Hessian of the loss times d subject to the norm of d being at most 1 under an arbitrary norm, generalizing the largest-eigenvalue sharpness used for vanilla gradient descent (Definition 2.2).",
        "file": "pages/claim-2-def22-generalized-sharpness/page.md",
        "children": []
      },
      {
        "slug": "claim-3-thm51-quadratic-linear-convergence",
        "title": "Theorem 5.1 proves non-Euclidean gradient descent on quadratic objectives converges linearly when the step size eta < 2/S, where S is the generalized sharpness (Theorem 5.1).",
        "file": "pages/claim-3-thm51-quadratic-linear-convergence/page.md",
        "children": []
      },
      {
        "slug": "claim-4-thm52-quadratic-divergence",
        "title": "Theorem 5.2 proves divergence occurs from some initialization when eta > 2/S for non-Euclidean gradient descent on quadratics (Theorem 5.2).",
        "file": "pages/claim-4-thm52-quadratic-divergence/page.md",
        "children": []
      },
      {
        "slug": "claim-5-fig2-linf-descent-eos-above-threshold",
        "title": "Experiments show l-infinity-descent on CIFAR-10 and Tiny Shakespeare exhibits generalized sharpness stabilizing slightly above the 2/eta threshold, unlike vanilla GD which stabilizes at 2/eta (Figure 2).",
        "file": "pages/claim-5-fig2-linf-descent-eos-above-threshold/page.md",
        "children": []
      },
      {
        "slug": "claim-6-fig5-signgd-muon-normalized-eos",
        "title": "SignGD and Muon (without momentum) are shown to exhibit normalized sharpness converging near the 2/eta edge-of-stability threshold, extending the phenomenon to optimizers used in practice (Figure 5).",
        "file": "pages/claim-6-fig5-signgd-muon-normalized-eos/page.md",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "file": "pages/conclusion/page.md",
        "children": []
      }
    ]
  }
}