# Analytic Study of Families of Spurious Minima in Two-Layer ReLU Neural Networks: A Tale of Symmetry II

arXiv: https://arxiv.org/abs/2107.10370 · HTML: https://mlsanity.com/paper/2107.10370/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 70 references that are in the map.
- First posted on arXiv 21 Jul 2021.

## Builds on (most cited first, 30 of 37)

- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [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)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (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)
- [When Do Neural Networks Outperform Kernel Methods?](https://mlsanity.com/paper/2006.13409/) (2020)
- [Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity](https://mlsanity.com/paper/1602.05897/) (2016)
- [Convergence Analysis of Two-layer Neural Networks with ReLU Activation](https://mlsanity.com/paper/1705.09886/) (2017)
- [Dynamics of stochastic gradient descent for two-layer neural networks in the teacher-student setup](https://mlsanity.com/paper/1906.08632/) (2019)
- [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)
- [Universal Statistics of Fisher Information in Deep Neural Networks: Mean Field Approach](https://mlsanity.com/paper/1806.01316/) (2018)
- [Hessian-based Analysis of Large Batch Training and Robustness to Adversaries](https://mlsanity.com/paper/1802.08241/) (2018)
- [Towards Understanding Generalization of Deep Learning: Perspective of Loss Landscapes](https://mlsanity.com/paper/1706.10239/) (2017)
- [Learning One-hidden-layer Neural Networks with Landscape Design](https://mlsanity.com/paper/1711.00501/) (2017)
- [A Random Matrix Approach to Neural Networks](https://mlsanity.com/paper/1702.05419/) (2017)
- [Globally Optimal Gradient Descent for a ConvNet with Gaussian Inputs](https://mlsanity.com/paper/1702.07966/) (2017)
- [The committee machine: Computational to statistical gaps in learning a two-layers neural network](https://mlsanity.com/paper/1806.05451/) (2018)
- [An Analytical Formula of Population Gradient for two-layered ReLU network and its Applications in Convergence and Critical Point Analysis](https://mlsanity.com/paper/1703.00560/) (2017)
- [Optimization and Generalization of Shallow Neural Networks with Quadratic Activation Functions](https://mlsanity.com/paper/2006.15459/) (2020)
- [Gradient Descent Learns One-hidden-layer CNN: Don't be Afraid of Spurious Local Minima](https://mlsanity.com/paper/1712.00779/) (2017)
- [Distribution-Specific Hardness of Learning Neural Networks](https://mlsanity.com/paper/1609.01037/) (2016)
- [Analytic Characterization of the Hessian in Shallow ReLU Models: A Tale of Symmetry](https://mlsanity.com/paper/2008.01805/) (2020)
- [On Learnability via Gradient Method for Two-Layer ReLU Neural Networks in Teacher-Student Setting](https://mlsanity.com/paper/2106.06251/) (2021)

## Built on by (most cited first, 16 of 16)

- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [On the Effective Number of Linear Regions in Shallow Univariate ReLU Networks: Convergence Guarantees and Implicit Bias](https://mlsanity.com/paper/2205.09072/) (2022)
- [Annihilation of Spurious Minima in Two-Layer ReLU Networks](https://mlsanity.com/paper/2210.06088/) (2022)
- [Learning Gaussian Multi-Index Models with Gradient Flow: Time Complexity and Directional Convergence](https://mlsanity.com/paper/2411.08798/) (2024)
- [On the Omnipresence of Spurious Local Minima in Certain Neural Network Training Problems](https://mlsanity.com/paper/2202.12262/) (2022)
- [Should Under-parameterized Student Networks Copy or Average Teacher Weights?](https://mlsanity.com/paper/2311.01644/) (2023)
- [A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models](https://mlsanity.com/paper/2506.02269/) (2025)
- [Loss Landscape of Shallow ReLU-like Neural Networks: Stationary Points, Saddle Escape, and Network Embedding](https://mlsanity.com/paper/2402.05626/) (2024)
- [Magnitude and Angle Dynamics in Training Single ReLU Neurons](https://mlsanity.com/paper/2209.13394/) (2022)
- [Symmetry & Critical Points for Symmetric Tensor Decomposition Problems](https://mlsanity.com/paper/2306.07886/) (2023)
- [Flat Channels to Infinity in Neural Loss Landscapes](https://mlsanity.com/paper/2506.14951/) (2025)
- [Hidden Minima in Two-Layer ReLU Networks](https://mlsanity.com/paper/2312.16819/) (2023)
- [Symmetry & Critical Points](https://mlsanity.com/paper/2408.14445/) (2024)
- [Nonlinear Dynamics In Optimization Landscape of Shallow Neural Networks with Tunable Leaky ReLU](https://mlsanity.com/paper/2510.25060/) (2025)
- [Explaining Near-Zero Hessian Eigenvalues Through Approximate Symmetries in Neural Networks](https://mlsanity.com/paper/2607.07845/) (2026)
- [Removing spurious minima for planar features by skip connections](https://mlsanity.com/paper/2610.01728/) (2026)

## Abstract (arXiv)

We study the optimization problem associated with fitting two-layer ReLU neural networks with respect to the squared loss, where labels are generated by a target network. We make use of the rich symmetry structure to develop a novel set of tools for studying families of spurious minima. In contrast to existing approaches which operate in limiting regimes, our technique directly addresses the nonconvex loss landscape for a finite number of inputs d and neurons k, and provides analytic, rather than heuristic, information. In particular, we derive analytic estimates for the loss at different minima, and prove that modulo O(d-1/2)-terms the Hessian spectrum concentrates near small positive constants, with the exception of Theta(d) eigenvalues which grow linearly with~d. We further show that the Hessian spectrum at global and spurious minima coincide to O(d-1/2)-order, thus challenging our ability to argue about statistical generalization through local curvature. Lastly, our technique provides the exact fractional dimensionality at which families of critical points turn from saddles into spurious minima. This makes possible the study of the creation and the annihilation of spurious minima using powerful tools from equivariant bifurcation theory.