# Efficient approaches for escaping higher order saddle points in non-convex optimization

arXiv: https://arxiv.org/abs/1602.05908 · HTML: https://mlsanity.com/paper/1602.05908/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2016.

## Built on by (most cited first, 23 of 23)

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- [Boundary Conditions for Linear Exit Time Gradient Trajectories Around Saddle Points: Analysis and Algorithm](https://mlsanity.com/paper/2101.02625/) (2021)
- [Escaping Saddle Points with Stochastically Controlled Stochastic Gradient Methods](https://mlsanity.com/paper/2103.04413/) (2021)
- [HOUDINI: Escaping from Moderately Constrained Saddles](https://mlsanity.com/paper/2205.13753/) (2022)
- [Faster Riemannian Newton-type Optimization by Subsampling and Cubic Regularization](https://mlsanity.com/paper/2302.11076/) (2023)
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- [Generalized Random Direction Newton Algorithms for Stochastic Optimization](https://mlsanity.com/paper/2602.19893/) (2026)
- [Benign Loss Landscapes Can Coexist with Worst-Case Hardness](https://mlsanity.com/paper/2609.13057/) (2026)

## Abstract (arXiv)

Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima.