Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks
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Abstract (arXiv)
We study the gradient descent (GD) dynamics of a depth-2 linear neural network with a single input and output. We show that GD converges at an explicit linear rate to a global minimum of the training loss, even with a large stepsize -- about 2/textrmsharpness. It still converges for even larger stepsizes, but may do so very slowly. We also characterize the solution to which GD converges, which has lower norm and sharpness than the gradient flow solution. Our analysis reveals a trade off between the speed of convergence and the magnitude of implicit regularization. This sheds light on the benefits of training at the ``Edge of Stability'', which induces additional regularization by delaying convergence and may have implications for training more complex models.
Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 16 of its 23 references that are in the map. First posted on arXiv 15 Jan 2025.
Builds on
16 of 23 references.
References · 16
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- 2021Gradient Descent on Neural Networks Typically Occurs at the Edge of Stabilitycited by 413
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- 2022Learning threshold neurons via the "edge of stability"cited by 50
- 2021Large Learning Rate Tames Homogeneity: Convergence and Balancing Effectcited by 48
4 more
- 2022Beyond the Edge of Stability via Two-step Gradient Updatescited by 38
- 2021Convergence of gradient descent for learning linear neural networkscited by 22
- 2019The Effect of Network Width on Stochastic Gradient Descent and Generalization: an Empirical Studycited by 16
- 2024Three Mechanisms of Feature Learning in a Linear Networkcited by 3
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Citing papers per year: 2026: 2.
2 papers.