# Noether: The More Things Change, the More Stay the Same

arXiv: https://arxiv.org/abs/2104.05508 · HTML: https://mlsanity.com/paper/2104.05508/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 50 references that are in the map.
- First posted on arXiv 12 Apr 2021.

## Builds on (most cited first, 30 of 40)

- [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)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Uniform convergence may be unable to explain generalization in deep learning](https://mlsanity.com/paper/1902.04742/) (2019)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [When Do Neural Networks Outperform Kernel Methods?](https://mlsanity.com/paper/2006.13409/) (2020)
- [Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1811.08888/) (2018)
- [Identity Matters in Deep Learning](https://mlsanity.com/paper/1611.04231/) (2016)
- [Spurious Local Minima are Common in Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/1712.08968/) (2017)
- [On the Power and Limitations of Random Features for Understanding Neural Networks](https://mlsanity.com/paper/1904.00687/) (2019)
- [Topology and Geometry of Half-Rectified Network Optimization](https://mlsanity.com/paper/1611.01540/) (2016)

## Built on by (most cited first, 19 of 19)

- [Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances](https://mlsanity.com/paper/2105.12221/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Abide by the Law and Follow the Flow: Conservation Laws for Gradient Flows](https://mlsanity.com/paper/2307.00144/) (2023)
- [Symmetry in Neural Network Parameter Spaces](https://mlsanity.com/paper/2506.13018/) (2025)
- [Symmetries, flat minima, and the conserved quantities of gradient flow](https://mlsanity.com/paper/2210.17216/) (2022)
- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)
- [Symmetry Teleportation for Accelerated Optimization](https://mlsanity.com/paper/2205.10637/) (2022)
- [Keep the Momentum: Conservation Laws beyond Euclidean Gradient Flows](https://mlsanity.com/paper/2405.12888/) (2024)
- [TeleSparse: Practical Privacy-Preserving Verification of Deep Neural Networks](https://mlsanity.com/paper/2504.19274/) (2025)
- [On the symmetries in the dynamics of wide two-layer neural networks](https://mlsanity.com/paper/2211.08771/) (2022)
- [Topological obstruction to the training of shallow ReLU neural networks](https://mlsanity.com/paper/2410.14837/) (2024)
- [Topology and Geometry of the Learning Space of ReLU Networks: Connectivity and Singularities](https://mlsanity.com/paper/2602.00693/) (2026)
- [The Symmetries of Three-Layer ReLU Networks](https://mlsanity.com/paper/2605.18319/) (2026)
- [WARP: Weight Teleportation for Attack-Resilient Unlearning Protocols](https://mlsanity.com/paper/2512.00272/) (2025)
- [Conservation Laws from Data Symmetry in Neural Networks](https://mlsanity.com/paper/2606.10913/) (2026)
- [Complexity from Adaptive-Symmetries Breaking: Global Minima in the Statistical Mechanics of Deep Neural Networks](https://mlsanity.com/paper/2201.07934/) (2022)
- [Sequencing the Neurome: Towards Scalable Exact Parameter Reconstruction of Black-Box Neural Networks](https://mlsanity.com/paper/2409.19138/) (2024)
- [An Equivariance Toolbox for Learning Dynamics](https://mlsanity.com/paper/2512.21447/) (2025)
- [On Parameter Symmetries and Conservation Laws in Gradient Flow](https://mlsanity.com/paper/2609.34549/) (2026)

## Abstract (arXiv)

Symmetries have proven to be important ingredients in the analysis of neural networks. So far their use has mostly been implicit or seemingly coincidental. We undertake a systematic study of the role that symmetry plays. In particular, we clarify how symmetry interacts with the learning algorithm. The key ingredient in our study is played by Noether's celebrated theorem which, informally speaking, states that symmetry leads to conserved quantities (e.g., conservation of energy or conservation of momentum). In the realm of neural networks under gradient descent, model symmetries imply restrictions on the gradient path. E.g., we show that symmetry of activation functions leads to boundedness of weight matrices, for the specific case of linear activations it leads to balance equations of consecutive layers, data augmentation leads to gradient paths that have "momentum"-type restrictions, and time symmetry leads to a version of the Neural Tangent Kernel. Symmetry alone does not specify the optimization path, but the more symmetries are contained in the model the more restrictions are imposed on the path. Since symmetry also implies over-parametrization, this in effect implies that some part of this over-parametrization is cancelled out by the existence of the conserved quantities. Symmetry can therefore be thought of as one further important tool in understanding the performance of neural networks under gradient descent.