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arXiv · 2509.25351

Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region

Shuang Liang·Guido Montúfar
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Abstract

We examine gradient descent in matrix factorization and show that under large step sizes the parameter space develops a fractal structure. We derive the exact critical step size for convergence in scalar-vector factorization and show that near criticality the selected minimizer depends sensitively on the initialization. Moreover, we show that adding regularization amplifies this sensitivity, generating a fractal boundary between initializations that converge and those that diverge. The analysis extends to general matrix factorization with orthogonal initialization. Our findings reveal that near-critical step sizes induce a chaotic regime of gradient descent where the training outcome is unpredictable and there are no simple implicit biases, such as towards balancedness, minimum norm, or flatness.

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Shuang Liang, Guido Montúfar. 2026-02-08. Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region. https://arxiv.org/abs/2509.25351

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