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arXiv · 2507.12709

From SGD to Spectra: A Theory of Neural Network Weight Dynamics

Brian Richard Olsen·Sam Fatehmanesh·Frank Xiao·Adarsh Kumarappan·Anirudh Gajula
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Abstract

Deep neural networks have revolutionized machine learning, yet their training dynamics remain theoretically unclear-we develop a continuous-time, matrix-valued stochastic differential equation (SDE) framework that rigorously connects the microscopic dynamics of SGD to the macroscopic evolution of singular-value spectra in weight matrices. We derive exact SDEs showing that squared singular values follow Dyson Brownian motion with eigenvalue repulsion, and characterize stationary distributions as gamma-type densities with power-law tails, providing the first theoretical explanation for the empirically observed 'bulk+tail' spectral structure in trained networks. Through controlled experiments on transformer and MLP architectures, we validate our theoretical predictions and demonstrate quantitative agreement between SDE-based forecasts and observed spectral evolution, providing a rigorous foundation for understanding why deep learning works.

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Brian Richard Olsen, Sam Fatehmanesh, Frank Xiao, Adarsh Kumarappan, Anirudh Gajula. 2025-07-17. From SGD to Spectra: A Theory of Neural Network Weight Dynamics. https://arxiv.org/abs/2507.12709

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