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arXiv · 2609.36271

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

Thomas Dybdahl Ahle·Yaroslav Bulatov·Christopher De Sa·Christopher Ré
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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.

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Thomas Dybdahl Ahle, Yaroslav Bulatov, Christopher De Sa, Christopher Ré. 2026-09-28. Stochastic Optimization Under Power-Law Spectra: Tight Bounds and Shuffling Analysis. https://arxiv.org/abs/2609.36271

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