# Understanding Sharpness Dynamics in NN Training with a Minimalist Example: The Effects of Dataset Difficulty, Depth, Stochasticity, and More

arXiv: https://arxiv.org/abs/2506.06940 · HTML: https://mlsanity.com/paper/2506.06940/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 37 references that are in the map.
- First posted on arXiv 7 Jun 2025.

## Builds on (most cited first, 28 of 28)

- [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)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Towards Explaining the Regularization Effect of Initial Large Learning Rate in Training Neural Networks](https://mlsanity.com/paper/1907.04595/) (2019)
- [Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2209.15594/) (2022)
- [The Break-Even Point on Optimization Trajectories of Deep Neural Networks](https://mlsanity.com/paper/2002.09572/) (2020)
- [Understanding Gradient Descent on Edge of Stability in Deep Learning](https://mlsanity.com/paper/2205.09745/) (2022)
- [Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity](https://mlsanity.com/paper/2106.09524/) (2021)
- [A Modern Look at the Relationship between Sharpness and Generalization](https://mlsanity.com/paper/2302.07011/) (2023)
- [On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length](https://mlsanity.com/paper/1807.05031/) (2018)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [Understanding Optimization in Deep Learning with Central Flows](https://mlsanity.com/paper/2410.24206/) (2024)
- [Analyzing Sharpness along GD Trajectory: Progressive Sharpening and Edge of Stability](https://mlsanity.com/paper/2207.12678/) (2022)
- [Understanding Edge-of-Stability Training Dynamics with a Minimalist Example](https://mlsanity.com/paper/2210.03294/) (2022)
- [Learning threshold neurons via the "edge of stability"](https://mlsanity.com/paper/2212.07469/) (2022)
- [Implicit Bias of Gradient Descent for Logistic Regression at the Edge of Stability](https://mlsanity.com/paper/2305.11788/) (2023)
- [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)
- [Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory](https://mlsanity.com/paper/2307.04204/) (2023)
- [Universal Sharpness Dynamics in Neural Network Training: Fixed Point Analysis, Edge of Stability, and Route to Chaos](https://mlsanity.com/paper/2311.02076/) (2023)
- [Gradient Descent Monotonically Decreases the Sharpness of Gradient Flow Solutions in Scalar Networks and Beyond](https://mlsanity.com/paper/2305.13064/) (2023)
- [High dimensional analysis reveals conservative sharpening and a stochastic edge of stability](https://mlsanity.com/paper/2404.19261/) (2024)
- [Deep linear networks for regression are implicitly regularized towards flat minima](https://mlsanity.com/paper/2405.13456/) (2024)
- [Outliers with Opposing Signals Have an Outsized Effect on Neural Network Optimization](https://mlsanity.com/paper/2311.04163/) (2023)

## Built on by (most cited first, 5 of 5)

- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [Dynamics of Gradient Descent with Large Step Size Near a Manifold of Flat Minima](https://mlsanity.com/paper/2607.08380/) (2026)
- [An Equivariance Toolbox for Learning Dynamics](https://mlsanity.com/paper/2512.21447/) (2025)
- [Zeroth-Order Optimization at the Edge of Stability](https://mlsanity.com/paper/2604.14669/) (2026)
- [The Multiple Timescales of Gradient Descent on the Edge of Stability: A Perturbative Derivation of the Central Flow](https://mlsanity.com/paper/2609.01034/) (2026)

## Abstract (arXiv)

When training deep neural networks with gradient descent, sharpness often increases -- a phenomenon known as progressive sharpening -- before saturating at the edge of stability. Although commonly observed in practice, the underlying mechanisms behind progressive sharpening remain poorly understood. In this work, we study this phenomenon using a minimalist model: a deep linear network with a single neuron per layer. We show that this simple model effectively captures the sharpness dynamics observed in recent empirical studies, offering a simple testbed to better understand neural network training. Moreover, we theoretically analyze how dataset properties, network depth, stochasticity of optimizers, and step size affect the degree of progressive sharpening in the minimalist model. We then empirically demonstrate how these theoretical insights extend to practical scenarios. This study offers a deeper understanding of sharpness dynamics in neural network training, highlighting the interplay between depth, training data, and optimizers.