| { | |
| "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": [] | |
| } | |
| ] | |
| } | |
| } |