# Tight Transition Time Bounds for Separable Logistic Regression at the Edge of Stability

arXiv: https://arxiv.org/abs/2610.01459 · HTML: https://mlsanity.com/paper/2610.01459/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 40 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Builds on (most cited first, 7 of 7)

- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [Adaptive Gradient Methods at the Edge of Stability](https://mlsanity.com/paper/2207.14484/) (2022)
- [Understanding Optimization in Deep Learning with Central Flows](https://mlsanity.com/paper/2410.24206/) (2024)
- [Edge of Stochastic Stability: Revisiting the Edge of Stability for SGD](https://mlsanity.com/paper/2412.20553/) (2024)
- [Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes](https://mlsanity.com/paper/2406.05033/) (2024)
- [A Defense of the Quadratic Model](https://mlsanity.com/paper/2607.21716/) (2026)
- [Gradient Descent on Logistic Regression: Do Large Step-Sizes Work with Data on the Sphere?](https://mlsanity.com/paper/2507.11228/) (2025)

## Abstract (arXiv)

We study logistic regression on linearly separable data under gradient descent with a large constant stepsize η. Such dynamics may exhibit a characteristic Edge of Stability phenomenon, in which the loss initially oscillates before transitioning to a stable phase of monotone decrease. Existing work provides a tight Theta(1) bound in dimension d=2 as η → ∞ and conjectures a bound independent of η in arbitrary dimensions d≥ 2. In this paper, we disprove this conjecture by showing that, for every fixed sample size n≥ 2 and sufficiently small margin γ, the worst-case transition time is Theta!left((logη)^minn-2,d-2right) uniformly over d≥2. The key challenge in establishing a tight bound is that the sample contributing most strongly to the gradient can change repeatedly across iterations. To address this issue, we control such changes by induction on dimension and sample size, and construct matching hard instances.