Non-Euclidean Gradient Descent Has Many Edges of Stability

Peter Zaika, Zakhar Shumaylov

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…