Mathematics > Optimization and Control
arXiv:2502.11152 (math)
[Submitted on 16 Feb 2025 (v1), last revised 18 Feb 2025 (this version, v2)]
Title:Error Bound Analysis for the Regularized Loss of Deep Linear Neural Networks
View a PDF of the paper titled Error Bound Analysis for the Regularized Loss of Deep Linear Neural Networks, by Po Chen and 2 other authors
View PDFAbstract:The optimization foundations of deep linear networks have received significant attention lately. However, due to the non-convexity and hierarchical structure, analyzing the regularized loss of deep linear networks remains a challenging task. In this work, we study the local geometric landscape of the regularized squared loss of deep linear networks, providing a deeper understanding of its optimization properties. Specifically, we characterize the critical point set and establish an error-bound property for all critical points under mild conditions. Notably, we identify the sufficient and necessary conditions under which the error bound holds. To support our theoretical findings, we conduct numerical experiments demonstrating that gradient descent exhibits linear convergence when optimizing the regularized loss of deep linear networks.
| Comments: | 55 pages, 2 figures |
| Subjects: | Optimization and Control (math.OC); Machine Learning (cs.LG) |
| MSC classes: | 90C26, 68T07, 65K10 |
| Cite as: | arXiv:2502.11152 [math.OC] |
| (orarXiv:2502.11152v2 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2502.11152 arXiv-issued DOI via DataCite |
Submission history
From: Peng Wang [view email][v1] Sun, 16 Feb 2025 14:53:52 UTC (5,478 KB)
[v2] Tue, 18 Feb 2025 04:13:36 UTC (729 KB)
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View a PDF of the paper titled Error Bound Analysis for the Regularized Loss of Deep Linear Neural Networks, by Po Chen and 2 other authors
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