Precise Dynamics of Diagonal Linear Networks: A Unifying Analysis by Dynamical Mean-Field Theory
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917337013157888 |
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| author | Nishiyama, Sota Imaizumi, Masaaki |
| author_facet | Nishiyama, Sota Imaizumi, Masaaki |
| contents | Diagonal linear networks (DLNs) are a tractable model that captures several nontrivial behaviors in neural network training, such as initialization-dependent solutions and incremental learning. These phenomena are typically studied in isolation, leaving the overall dynamics insufficiently understood. In this work, we present a unified analysis of various phenomena in the gradient flow dynamics of DLNs. Using Dynamical Mean-Field Theory (DMFT), we derive a low-dimensional effective process that captures the asymptotic gradient flow dynamics in high dimensions. Analyzing this effective process yields new insights into DLN dynamics, including loss convergence rates and their trade-off with generalization, and systematically reproduces many of the previously observed phenomena. These findings deepen our understanding of DLNs and demonstrate the effectiveness of the DMFT approach in analyzing high-dimensional learning dynamics of neural networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Precise Dynamics of Diagonal Linear Networks: A Unifying Analysis by Dynamical Mean-Field Theory Nishiyama, Sota Imaizumi, Masaaki Machine Learning Disordered Systems and Neural Networks Diagonal linear networks (DLNs) are a tractable model that captures several nontrivial behaviors in neural network training, such as initialization-dependent solutions and incremental learning. These phenomena are typically studied in isolation, leaving the overall dynamics insufficiently understood. In this work, we present a unified analysis of various phenomena in the gradient flow dynamics of DLNs. Using Dynamical Mean-Field Theory (DMFT), we derive a low-dimensional effective process that captures the asymptotic gradient flow dynamics in high dimensions. Analyzing this effective process yields new insights into DLN dynamics, including loss convergence rates and their trade-off with generalization, and systematically reproduces many of the previously observed phenomena. These findings deepen our understanding of DLNs and demonstrate the effectiveness of the DMFT approach in analyzing high-dimensional learning dynamics of neural networks. |
| title | Precise Dynamics of Diagonal Linear Networks: A Unifying Analysis by Dynamical Mean-Field Theory |
| topic | Machine Learning Disordered Systems and Neural Networks |
| url | https://arxiv.org/abs/2510.01930 |