Non-Euclidean Gradient Descent Has Many Edges of Stability
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 intermedi...
| Authors | Peter Zaika, Zakhar Shumaylov |
|---|---|
| Primary category | math.OC |
| Published | 10 October 2026 |
| Last updated at source | 10 October 2026 |
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