Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912886393143296 |
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| author | Liang, Shuang Montúfar, Guido |
| author_facet | Liang, Shuang Montúfar, Guido |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25351 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region Liang, Shuang Montúfar, Guido Machine Learning 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. |
| title | Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2509.25351 |