High-Probability Guarantees for SGD under $\beta$-Heavy-Tailed Gradient Noise

arXiv:2609.32195v1 Announce Type: cross Abstract: Stochastic gradient descent (SGD) is widely used to train machine learning models, but subsampling the training data introduces noise into its updates. The strength and applicability of high-probability guarantees therefore depend critically on how the tails of gradient noise are modeled. Reports of heavy-tailed gradient noise in deep learning motivate relaxing the bounded-noise and sub-Gaussian assumptions commonly used in high-probability analyses of SGD. We use Young functions from Orlicz space theory to describe noise tails in a common framework. We model SGD gradient noise by adopting a Young function that preserves the finiteness of all polynomial moments while allowing tails heavier than sub-Weibull, including lognormal distributions. The resulting class is called $\beta$-heavy-tailed, with $\beta$ controlling the tail heaviness. We establish concentration inequalities for $\beta$-heavy-tailed noise and combine them with a uniform bound on the difference between empirical and population gradients along the SGD trajectory to obtain high-probability bounds on optimization and population-risk stationarity for smooth nonconvex losses under trajectory assumptions. The bounds are not restricted to a particular learning-rate decay rule and make explicit the effects of noise tails and learning-rate schedules. Under the Polyak-{\L}ojasiewicz condition, we bound the risk at the last iterate. We also analyze SGD with gradient clipping under the $\beta$-heavy-tailed noise model.

Abstract:Stochastic gradient descent (SGD) is widely used to train machine learning models, but subsampling the training data introduces noise into its updates. The strength and applicability of high-probability guarantees therefore depend critically on how the tails of gradient noise are modeled. Reports of heavy-tailed gradient noise in deep learning motivate relaxing the bounded-noise and sub-Gaussian assumptions commonly used in high-probability analyses of SGD. We use Young functions from Orlicz space theory to describe noise tails in a common framework. We model SGD gradient noise by adopting a Young function that preserves the finiteness of all polynomial moments while allowing tails heavier than sub-Weibull, including lognormal distributions. The resulting class is called $\beta$-heavy-tailed, with $\beta$ controlling the tail heaviness. We establish concentration inequalities for $\beta$-heavy-tailed noise and combine them with a uniform bound on the difference between empirical and population gradients along the SGD trajectory to obtain high-probability bounds on optimization and population-risk stationarity for smooth nonconvex losses under trajectory assumptions. The bounds are not restricted to a particular learning-rate decay rule and make explicit the effects of noise tails and learning-rate schedules. Under the Polyak-Łojasiewicz condition, we bound the risk at the last iterate. We also analyze SGD with gradient clipping under the $\beta$-heavy-tailed noise model.

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

arXiv:2609.32195 [stat.ML]

(or arXiv:2609.32195v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.32195 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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High-Probability Guarantees for SGD under $\beta$-Heavy-Tailed Gradient Noise

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