# Gradient Descent Monotonically Decreases the Sharpness of Gradient Flow Solutions in Scalar Networks and Beyond

arXiv: https://arxiv.org/abs/2305.13064 · HTML: https://mlsanity.com/paper/2305.13064/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 35 references that are in the map.
- First posted on arXiv 22 May 2023.

## Builds on (most cited first, 23 of 23)

- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [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)
- [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)
- [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)
- [Adaptive Gradient Methods at the Edge of Stability](https://mlsanity.com/paper/2207.14484/) (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)
- [On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent](https://mlsanity.com/paper/2102.09769/) (2021)
- [The Implicit Bias of Depth: How Incremental Learning Drives Generalization](https://mlsanity.com/paper/1909.12051/) (2019)
- [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)
- [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)
- [Continuous vs. Discrete Optimization of Deep Neural Networks](https://mlsanity.com/paper/2107.06608/) (2021)

## Built on by (most cited first, 21 of 21)

- [Does SGD really happen in tiny subspaces?](https://mlsanity.com/paper/2405.16002/) (2024)
- [Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency](https://mlsanity.com/paper/2402.15926/) (2024)
- [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)
- [From Stability to Chaos: Analyzing Gradient Descent Dynamics in Quadratic Regression](https://mlsanity.com/paper/2310.01687/) (2023)
- [Large Stepsize Gradient Descent for Non-Homogeneous Two-Layer Networks: Margin Improvement and Fast Optimization](https://mlsanity.com/paper/2406.08654/) (2024)
- [Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes](https://mlsanity.com/paper/2406.05033/) (2024)
- [Stable Minima Cannot Overfit in Univariate ReLU Networks: Generalization by Large Step Sizes](https://mlsanity.com/paper/2406.06838/) (2024)
- [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)
- [Understanding Sharpness Dynamics in NN Training with a Minimalist Example: The Effects of Dataset Difficulty, Depth, Stochasticity, and More](https://mlsanity.com/paper/2506.06940/) (2025)
- [A Minimalist Example of Edge-of-Stability and Progressive Sharpening](https://mlsanity.com/paper/2503.02809/) (2025)
- [Gradient Descent with Polyak's Momentum Finds Flatter Minima via Large Catapults](https://mlsanity.com/paper/2311.15051/) (2023)
- [On the Stability of Nonlinear Dynamics in GD and SGD: Beyond Quadratic Potentials](https://mlsanity.com/paper/2602.14789/) (2026)
- [Variational Learning Finds Flatter Solutions at the Edge of Stability](https://mlsanity.com/paper/2506.12903/) (2025)
- [SGD at the Edge of Stability: The Stochastic Sharpness Gap](https://mlsanity.com/paper/2604.21016/) (2026)
- [The Road Taken: The Role of Optimizers at the Edge of Stability](https://mlsanity.com/paper/2608.18415/) (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)
- [Mind the Drift: Diagonal Linear Networks Under Large Learning Rates](https://mlsanity.com/paper/2610.06120/) (2026)

## Abstract (arXiv)

Recent research shows that when Gradient Descent (GD) is applied to neural networks, the loss almost never decreases monotonically. Instead, the loss oscillates as gradient descent converges to its ''Edge of Stability'' (EoS). Here, we find a quantity that does decrease monotonically throughout GD training: the sharpness attained by the gradient flow solution (GFS)-the solution that would be obtained if, from now until convergence, we train with an infinitesimal step size. Theoretically, we analyze scalar neural networks with the squared loss, perhaps the simplest setting where the EoS phenomena still occur. In this model, we prove that the GFS sharpness decreases monotonically. Using this result, we characterize settings where GD provably converges to the EoS in scalar networks. Empirically, we show that GD monotonically decreases the GFS sharpness in a squared regression model as well as practical neural network architectures.