# A Minimalist Example of Edge-of-Stability and Progressive Sharpening

arXiv: https://arxiv.org/abs/2503.02809 · HTML: https://mlsanity.com/paper/2503.02809/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 25 references that are in the map.
- First posted on arXiv 4 Mar 2025.

## Builds on (most cited first, 19 of 19)

- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [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)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [Understanding the Generalization Benefit of Normalization Layers: Sharpness Reduction](https://mlsanity.com/paper/2206.07085/) (2022)
- [A Walk with SGD](https://mlsanity.com/paper/1802.08770/) (2018)
- [Theoretical Analysis of Auto Rate-Tuning by Batch Normalization](https://mlsanity.com/paper/1812.03981/) (2018)
- [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)
- [Large Learning Rate Tames Homogeneity: Convergence and Balancing Effect](https://mlsanity.com/paper/2110.03677/) (2021)
- [Second-order regression models exhibit progressive sharpening to the edge of stability](https://mlsanity.com/paper/2210.04860/) (2022)
- [Beyond the Edge of Stability via Two-step Gradient Updates](https://mlsanity.com/paper/2206.04172/) (2022)
- [Robust Training of Neural Networks Using Scale Invariant Architectures](https://mlsanity.com/paper/2202.00980/) (2022)
- [Good regularity creates large learning rate implicit biases: edge of stability, balancing, and catapult](https://mlsanity.com/paper/2310.17087/) (2023)
- [Gradient Descent Monotonically Decreases the Sharpness of Gradient Flow Solutions in Scalar Networks and Beyond](https://mlsanity.com/paper/2305.13064/) (2023)
- [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, 4 of 4)

- [Convergence Rates for Gradient Descent on the Edge of Stability in Overparametrised Least Squares](https://mlsanity.com/paper/2510.17506/) (2025)
- [Dynamics of Gradient Descent with Large Step Size Near a Manifold of Flat Minima](https://mlsanity.com/paper/2607.08380/) (2026)
- [Centre manifold theorem for maps along manifolds of fixed points](https://mlsanity.com/paper/2604.18202/) (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)

Recent advances in deep learning optimization have unveiled two intriguing phenomena under large learning rates: Edge of Stability (EoS) and Progressive Sharpening (PS), challenging classical Gradient Descent (GD) analyses. Current research approaches, using either generalist frameworks or minimalist examples, face significant limitations in explaining these phenomena. This paper advances the minimalist approach by introducing a two-layer network with a two-dimensional input, where one dimension is relevant to the response and the other is irrelevant. Through this model, we rigorously prove the existence of progressive sharpening and self-stabilization under large learning rates, and establish non-asymptotic analysis of the training dynamics and sharpness along the entire GD trajectory. Besides, we connect our minimalist example to existing works by reconciling the existence of a well-behaved ``stable set" between minimalist and generalist analyses, and extending the analysis of Gradient Flow Solution sharpness to our two-dimensional input scenario. These findings provide new insights into the EoS phenomenon from both parameter and input data distribution perspectives, potentially informing more effective optimization strategies in deep learning practice.