# Learning threshold neurons via the "edge of stability"

arXiv: https://arxiv.org/abs/2212.07469 · HTML: https://mlsanity.com/paper/2212.07469/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 62 references that are in the map.
- First posted on arXiv 14 Dec 2022.

## Builds on (most cited first, 30 of 35)

- [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)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [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)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Towards moderate overparameterization: global convergence guarantees for training shallow neural networks](https://mlsanity.com/paper/1902.04674/) (2019)
- [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)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [Understanding Gradient Descent on Edge of Stability in Deep Learning](https://mlsanity.com/paper/2205.09745/) (2022)
- [Feature Purification: How Adversarial Training Performs Robust Deep Learning](https://mlsanity.com/paper/2005.10190/) (2020)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [Unveiling Transformers with LEGO: a synthetic reasoning task](https://mlsanity.com/paper/2206.04301/) (2022)
- [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 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)
- [SGD with Large Step Sizes Learns Sparse Features](https://mlsanity.com/paper/2210.05337/) (2022)
- [Analyzing Sharpness along GD Trajectory: Progressive Sharpening and Edge of Stability](https://mlsanity.com/paper/2207.12678/) (2022)
- [Catastrophic Fisher Explosion: Early Phase Fisher Matrix Impacts Generalization](https://mlsanity.com/paper/2012.14193/) (2020)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Understanding Edge-of-Stability Training Dynamics with a Minimalist Example](https://mlsanity.com/paper/2210.03294/) (2022)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Large Learning Rate Tames Homogeneity: Convergence and Balancing Effect](https://mlsanity.com/paper/2110.03677/) (2021)

## Built on by (most cited first, 30 of 51)

- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [Linear attention is (maybe) all you need (to understand transformer optimization)](https://mlsanity.com/paper/2310.01082/) (2023)
- [Implicit Bias of Gradient Descent for Logistic Regression at the Edge of Stability](https://mlsanity.com/paper/2305.11788/) (2023)
- [(S)GD over Diagonal Linear Networks: Implicit Regularisation, Large Stepsizes and Edge of Stability](https://mlsanity.com/paper/2302.08982/) (2023)
- [Does SGD really happen in tiny subspaces?](https://mlsanity.com/paper/2405.16002/) (2024)
- [Why Does Sharpness-Aware Minimization Generalize Better Than SGD?](https://mlsanity.com/paper/2310.07269/) (2023)
- [The Crucial Role of Normalization in Sharpness-Aware Minimization](https://mlsanity.com/paper/2305.15287/) (2023)
- [Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory](https://mlsanity.com/paper/2307.04204/) (2023)
- [Good regularity creates large learning rate implicit biases: edge of stability, balancing, and catapult](https://mlsanity.com/paper/2310.17087/) (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)
- [When and Why Momentum Accelerates SGD:An Empirical Study](https://mlsanity.com/paper/2306.09000/) (2023)
- [Sharpness-Aware Minimization and the Edge of Stability](https://mlsanity.com/paper/2309.12488/) (2023)
- [From Stability to Chaos: Analyzing Gradient Descent Dynamics in Quadratic Regression](https://mlsanity.com/paper/2310.01687/) (2023)
- [Fast TRAC: A Parameter-Free Optimizer for Lifelong Reinforcement Learning](https://mlsanity.com/paper/2405.16642/) (2024)
- [Large Stepsize Gradient Descent for Non-Homogeneous Two-Layer Networks: Margin Improvement and Fast Optimization](https://mlsanity.com/paper/2406.08654/) (2024)
- [Understanding Multi-phase Optimization Dynamics and Rich Nonlinear Behaviors of ReLU Networks](https://mlsanity.com/paper/2305.12467/) (2023)
- [A Survey on Compositional Learning of AI Models: Theoretical and Experimental Practices](https://mlsanity.com/paper/2406.08787/) (2024)
- [Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes](https://mlsanity.com/paper/2406.05033/) (2024)
- [Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networks](https://mlsanity.com/paper/2506.06489/) (2025)
- [How to escape sharp minima with random perturbations](https://mlsanity.com/paper/2305.15659/) (2023)
- [Convergence of Alternating Gradient Descent for Matrix Factorization](https://mlsanity.com/paper/2305.06927/) (2023)
- [Stable Minima Cannot Overfit in Univariate ReLU Networks: Generalization by Large Step Sizes](https://mlsanity.com/paper/2406.06838/) (2024)
- [Implicit Regularization of Sharpness-Aware Minimization for Scale-Invariant Problems](https://mlsanity.com/paper/2410.14802/) (2024)
- [Through the River: Understanding the Benefit of Schedule-Free Methods for Language Model Training](https://mlsanity.com/paper/2507.09846/) (2025)
- [Rod Flow: A Continuous-Time Model for Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2602.01480/) (2026)
- [Minimax Optimal Convergence of Gradient Descent in Logistic Regression via Large and Adaptive Stepsizes](https://mlsanity.com/paper/2504.04105/) (2025)
- [Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region](https://mlsanity.com/paper/2509.25351/) (2025)
- [Where Do Large Learning Rates Lead Us?](https://mlsanity.com/paper/2410.22113/) (2024)
- [Zeroth-Order Optimization Finds Flat Minima](https://mlsanity.com/paper/2506.05454/) (2025)

## Abstract (arXiv)

Existing analyses of neural network training often operate under the unrealistic assumption of an extremely small learning rate. This lies in stark contrast to practical wisdom and empirical studies, such as the work of J. Cohen et al. (ICLR 2021), which exhibit startling new phenomena (the "edge of stability" or "unstable convergence") and potential benefits for generalization in the large learning rate regime. Despite a flurry of recent works on this topic, however, the latter effect is still poorly understood. In this paper, we take a step towards understanding genuinely non-convex training dynamics with large learning rates by performing a detailed analysis of gradient descent for simplified models of two-layer neural networks. For these models, we provably establish the edge of stability phenomenon and discover a sharp phase transition for the step size below which the neural network fails to learn "threshold-like" neurons (i.e., neurons with a non-zero first-layer bias). This elucidates one possible mechanism by which the edge of stability can in fact lead to better generalization, as threshold neurons are basic building blocks with useful inductive bias for many tasks.