Dispelling the Curse of Singularities in Neural Network Optimizations

• URL: http://arxiv.org/abs/2602.01308v1
• Date: Sun, 01 Feb 2026 16:09:06 GMT
• Title: Dispelling the Curse of Singularities in Neural Network Optimizations
• Authors: Hengjie Cao, Mengyi Chen, Yifeng Yang, Fang Dong, Ruijun Huang, Anrui Chen, Jixian Zhou, Mingzhi Dong, Yujiang Wang, Dongsheng Li, Wenyi Fang, Yuanyi Lin, Fan Wu, Li Shang,
• Abstract summary: We show that the gradient Frobenius norms are bounded by the top singular values of the weight matrices, and as training progresses, the mutually reinforcing growth of weight and representation singularities, relaxes these bounds, escalating the risk of sharp loss explosions.<n>To counter this, we propose Parametric Singularity Smoothing (PSS), a lightweight, flexible, effective method for smoothing the singular spectra of weight matrices.
• Score: 22.05217959662069
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This work investigates the optimization instability of deep neural networks from a less-explored yet insightful perspective: the emergence and amplification of singularities in the parametric space. Our analysis reveals that parametric singularities inevitably grow with gradient updates and further intensify alignment with representations, leading to increased singularities in the representation space. We show that the gradient Frobenius norms are bounded by the top singular values of the weight matrices, and as training progresses, the mutually reinforcing growth of weight and representation singularities, termed the curse of singularities, relaxes these bounds, escalating the risk of sharp loss explosions. To counter this, we propose Parametric Singularity Smoothing (PSS), a lightweight, flexible, and effective method for smoothing the singular spectra of weight matrices. Extensive experiments across diverse datasets, architectures, and optimizers demonstrate that PSS mitigates instability, restores trainability even after failure, and improves both training efficiency and generalization.

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