# There is a Singularity in the Loss Landscape

arXiv: https://arxiv.org/abs/2201.06964 · HTML: https://mlsanity.com/paper/2201.06964/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 26 references that are in the map.
- First posted on arXiv 12 Jan 2022.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Revisiting Unreasonable Effectiveness of Data in Deep Learning Era](https://mlsanity.com/paper/1707.02968/) (2017)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Gradient Descent Happens in a Tiny Subspace](https://mlsanity.com/paper/1812.04754/) (2018)
- [Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond](https://mlsanity.com/paper/1611.07476/) (2016)
- [Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians](https://mlsanity.com/paper/1901.08244/) (2019)
- [The Recurrent Neural Tangent Kernel](https://mlsanity.com/paper/2006.10246/) (2020)
- [Deep Learning Theory Review: An Optimal Control and Dynamical Systems Perspective](https://mlsanity.com/paper/1908.10920/) (2019)
- [A Neural Tangent Kernel Perspective of GANs](https://mlsanity.com/paper/2106.05566/) (2021)
- [Towards an Understanding of Residual Networks Using Neural Tangent Hierarchy (NTH)](https://mlsanity.com/paper/2007.03714/) (2020)

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

- [Rolling Ball Optimizer: Learning by ironing out loss landscape wrinkles](https://mlsanity.com/paper/2505.19527/) (2025)

## Abstract (arXiv)

Despite the widespread adoption of neural networks, their training dynamics remain poorly understood. We show experimentally that as the size of the dataset increases, a point forms where the magnitude of the gradient of the loss becomes unbounded. Gradient descent rapidly brings the network close to this singularity in parameter space, and further training takes place near it. This singularity explains a variety of phenomena recently observed in the Hessian of neural network loss functions, such as training on the edge of stability and the concentration of the gradient in a top subspace. Once the network approaches the singularity, the top subspace contributes little to learning, even though it constitutes the majority of the gradient.