Non-Euclidean Gradient Descent Operates at the Edge of Stability
The Edge of Stability (EoS) is a phenomenon where the sharpness (largest eigenvalue) of the Hessian approaches and then hovers near the stability threshold $2/η$ during gradient descent (GD) with step size $η$. Despite (apparently) violating classical smoothness assumptions, EoS has been widely observed in deep learning, but its theoretical foundations remain incomplete. We provide an interpretation of EoS through the lens of Directional Smoothness [Mishkin et al., 2024]. This interpretation naturally extends to non-Euclidean norms, which we use to define generalized sharpness under an arbitrary norm. Our generalized sharpness measure includes previously studied vanilla GD and preconditio...
| Authors | Rustem Islamov, Michael Crawshaw, Jeremy Cohen et al. |
|---|---|
| Primary category | cs.LG |
| Published | 5 March 2026 |
| Last updated at source | 28 June 2026 |
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