# Extending the step-size restriction for gradient descent to avoid strict saddle points

arXiv: https://arxiv.org/abs/1908.01753 · HTML: https://mlsanity.com/paper/1908.01753/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Aug 2019.

## Built on by (most cited first, 2 of 2)

- [On the ISS Property of the Gradient Flow for Single Hidden-Layer Neural Networks with Linear Activations](https://mlsanity.com/paper/2305.09904/) (2023)
- [Split-Merge: A Difference-based Approach for Dominant Eigenvalue Problem](https://mlsanity.com/paper/2501.15131/) (2025)

## Abstract (arXiv)

We provide larger step-size restrictions for which gradient descent based algorithms (almost surely) avoid strict saddle points. In particular, consider a twice differentiable (non-convex) objective function whose gradient has Lipschitz constant L and whose Hessian is well-behaved. We prove that the probability of initial conditions for gradient descent with step-size up to 2/L converging to a strict saddle point, given one uniformly random initialization, is zero. This extends previous results up to the sharp limit imposed by the convex case. In addition, the arguments hold in the case when a learning rate schedule is given, with either a continuous decaying rate or a piece-wise constant schedule.