Stochastic Optimization Under Power-Law Spectra: Tight Bounds and Shuffling Analysis

Abstract:Recent work has established that power-law spectral conditions on data enable tight convergence bounds for deterministic gradient descent, resolving the conflict between classical exponential bounds and observed power-law learning curves. In this work, we extend this result to the stochastic regime of high-dimensional machine learning. We provide two main contributions: (1) We generalize the power-law spectral theory to Stochastic Gradient Descent (SGD), showing that the same spectral exponents govern stochastic dynamics; (2) For the fundamental case of isotropic Gaussian data, we provide a precise analysis of data shuffling, deriving exact constants that prove Single Shuffle is strictly superior to Flip-Flop and IID sampling. Our results bridge the gap between abstract spectral theory and practical stochastic training choices, offering a unified picture of how data geometry drives optimization speed.

Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2609.36271 [cs.LG]

(or arXiv:2609.36271v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.36271 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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