# Efficiently Escaping Saddle Points under Generalized Smoothness via Self-Bounding Regularity

arXiv: https://arxiv.org/abs/2503.04712 · HTML: https://mlsanity.com/paper/2503.04712/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Mar 2025.

## Built on by (most cited first, 1 of 1)

- [On the Interaction of Batch Noise, Adaptivity, and Compression, under (L0,L1)-Smoothness: An SDE Approach](https://mlsanity.com/paper/2506.00181/) (2025)

## Abstract (arXiv)

We study the optimization of non-convex functions that are not necessarily smooth (gradient and/or Hessian are Lipschitz) using first order methods. Smoothness is a restrictive assumption in machine learning in both theory and practice, motivating significant recent work on finding first order stationary points of functions satisfying generalizations of smoothness with first order methods. We develop a novel framework that lets us systematically study the convergence of a large class of first-order optimization algorithms (which we call decrease procedures) under generalizations of smoothness. We instantiate our framework to analyze the convergence of first order optimization algorithms to first and textitsecond order stationary points under generalizations of smoothness. As a consequence, we establish the first convergence guarantees for first order methods to second order stationary points under generalizations of smoothness. We demonstrate that several canonical examples fall under our framework, and highlight practical implications.