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

• URL: http://arxiv.org/abs/2509.25351v2
• Date: Fri, 03 Oct 2025 00:33:57 GMT
• Title: Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region
• Abstract summary: We show that under large step sizes the parameter space develops a fractal structure in matrix factorization.<n>Our findings reveal that near-critical step sizes induce a chaotic regime of gradient descent where the long-term dynamics are unpredictable.
• Score: 16.679678607924007
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 long-term dynamics are unpredictable and there are no simple implicit biases, such as towards balancedness, minimum norm, or flatness.

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