Precise Dynamics of Diagonal Linear Networks: A Unifying Analysis by Dynamical Mean-Field Theory

• URL: http://arxiv.org/abs/2510.01930v1
• Date: Thu, 02 Oct 2025 11:47:36 GMT
• Title: Precise Dynamics of Diagonal Linear Networks: A Unifying Analysis by Dynamical Mean-Field Theory
• Authors: Sota Nishiyama, Masaaki Imaizumi,
• Abstract summary: Diagonal linear networks (DLNs) are a tractable model that captures several nontrivial behaviors in neural network training.<n>We present a unified analysis of various phenomena in the gradient flow dynamics of DLNs.
• Score: 6.2000582635449994
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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