# Quadratic models for understanding catapult dynamics of neural networks

arXiv: https://arxiv.org/abs/2205.11787 · HTML: https://mlsanity.com/paper/2205.11787/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 29 references that are in the map.
- First posted on arXiv 24 May 2022.

## Builds on (most cited first, 21 of 21)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [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)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [Feature Learning in Infinite-Width Neural Networks](https://mlsanity.com/paper/2011.14522/) (2020)
- [Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the Neural Tangent Kernel](https://mlsanity.com/paper/2010.15110/) (2020)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [An Improved Analysis of Training Over-parameterized Deep Neural Networks](https://mlsanity.com/paper/1906.04688/) (2019)
- [Polylogarithmic width suffices for gradient descent to achieve arbitrarily small test error with shallow ReLU networks](https://mlsanity.com/paper/1909.12292/) (2019)
- [Which Algorithmic Choices Matter at Which Batch Sizes? Insights From a Noisy Quadratic Model](https://mlsanity.com/paper/1907.04164/) (2019)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Dynamics of Deep Neural Networks and Neural Tangent Hierarchy](https://mlsanity.com/paper/1909.08156/) (2019)
- [The Interpolation Phase Transition in Neural Networks: Memorization and Generalization under Lazy Training](https://mlsanity.com/paper/2007.12826/) (2020)
- [Beyond Linearization: On Quadratic and Higher-Order Approximation of Wide Neural Networks](https://mlsanity.com/paper/1910.01619/) (2019)
- [What can linearized neural networks actually say about generalization?](https://mlsanity.com/paper/2106.06770/) (2021)
- [Gradient Dynamics of Shallow Univariate ReLU Networks](https://mlsanity.com/paper/1906.07842/) (2019)
- [Properties of the After Kernel](https://mlsanity.com/paper/2105.10585/) (2021)
- [Lyapunov Theory for Discrete Time Systems](https://mlsanity.com/paper/1809.05289/) (2018)
- [Increasing Depth Leads to U-Shaped Test Risk in Over-parameterized Convolutional Networks](https://mlsanity.com/paper/2010.09610/) (2020)

## Built on by (most cited first, 17 of 17)

- [Second-order regression models exhibit progressive sharpening to the edge of stability](https://mlsanity.com/paper/2210.04860/) (2022)
- [Catapults in SGD: spikes in the training loss and their impact on generalization through feature learning](https://mlsanity.com/paper/2306.04815/) (2023)
- [Mechanism of feature learning in deep fully connected networks and kernel machines that recursively learn features](https://mlsanity.com/paper/2212.13881/) (2022)
- [Universal Sharpness Dynamics in Neural Network Training: Fixed Point Analysis, Edge of Stability, and Route to Chaos](https://mlsanity.com/paper/2311.02076/) (2023)
- [Phase diagram of early training dynamics in deep neural networks: effect of the learning rate, depth, and width](https://mlsanity.com/paper/2302.12250/) (2023)
- [High dimensional analysis reveals conservative sharpening and a stochastic edge of stability](https://mlsanity.com/paper/2404.19261/) (2024)
- [Provable Multi-Task Representation Learning by Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/2307.06887/) (2023)
- [Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes](https://mlsanity.com/paper/2406.05033/) (2024)
- [FedP3: Federated Personalized and Privacy-friendly Network Pruning under Model Heterogeneity](https://mlsanity.com/paper/2404.09816/) (2024)
- [Benign Oscillation of Stochastic Gradient Descent with Large Learning Rates](https://mlsanity.com/paper/2310.17074/) (2023)
- [Closed-Form Training Dynamics Reveal Learned Features and Linear Structure in Word2Vec-like Models](https://mlsanity.com/paper/2502.09863/) (2025)
- [Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region](https://mlsanity.com/paper/2509.25351/) (2025)
- [Learning a Single Index Model from Anisotropic Data with vanilla Stochastic Gradient Descent](https://mlsanity.com/paper/2503.23642/) (2025)
- [Gradient Descent with Polyak's Momentum Finds Flatter Minima via Large Catapults](https://mlsanity.com/paper/2311.15051/) (2023)
- [A dynamic view of some anomalous phenomena in SGD](https://mlsanity.com/paper/2505.01751/) (2025)
- [Variational Learning Finds Flatter Solutions at the Edge of Stability](https://mlsanity.com/paper/2506.12903/) (2025)
- [Large Spikes in Stochastic Gradient Descent: A Large-Deviations View](https://mlsanity.com/paper/2603.10079/) (2026)

## Abstract (arXiv)

While neural networks can be approximated by linear models as their width increases, certain properties of wide neural networks cannot be captured by linear models. In this work we show that recently proposed Neural Quadratic Models can exhibit the "catapult phase" [Lewkowycz et al. 2020] that arises when training such models with large learning rates. We then empirically show that the behaviour of neural quadratic models parallels that of neural networks in generalization, especially in the catapult phase regime. Our analysis further demonstrates that quadratic models can be an effective tool for analysis of neural networks.