Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region
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.
Cite
@article{arxiv.2509.25351,
title = {Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region},
author = {Shuang Liang and Guido Montúfar},
journal= {arXiv preprint arXiv:2509.25351},
year = {2026}
}