# Are Saddles Good Enough for Deep Learning?

arXiv: https://arxiv.org/abs/1706.02052 · HTML: https://mlsanity.com/paper/1706.02052/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2017.

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

- [A Dynamics Theory of Implicit Regularization in Deep Low-Rank Matrix Factorization](https://mlsanity.com/paper/2212.14150/) (2022)
- [Benign Loss Landscapes Can Coexist with Worst-Case Hardness](https://mlsanity.com/paper/2609.13057/) (2026)

## Abstract (arXiv)

Recent years have seen a growing interest in understanding deep neural networks from an optimization perspective. It is understood now that converging to low-cost local minima is sufficient for such models to become effective in practice. However, in this work, we propose a new hypothesis based on recent theoretical findings and empirical studies that deep neural network models actually converge to saddle points with high degeneracy. Our findings from this work are new, and can have a significant impact on the development of gradient descent based methods for training deep networks. We validated our hypotheses using an extensive experimental evaluation on standard datasets such as MNIST and CIFAR-10, and also showed that recent efforts that attempt to escape saddles finally converge to saddles with high degeneracy, which we define as `good saddles'. We also verified the famous Wigner's Semicircle Law in our experimental results.