aiqbench analysis
This paper proves optimal last-iterate convergence rates in high probability for Clipped-SGD under heavy-tailed gradient noise, a setting where standard SGD can fail to converge. Its value is theoretical: it extends guarantees beyond the light-tailed noise assumptions that existing results rely on, which better matches conditions in practice. Readers training models with noisy or outlier-prone gradients should note that the analysis covers strongly convex problems, so its direct relevance to deep non-convex training remains an open question.
On the Last-Iterate Convergence of Clipped Gradient Methods
arXiv:2610.21026v1 Announce Type: cross Abstract: In stochastic (strongly) convex optimization, establishing last-iterate convergence guarantees for first-order methods has recently garnered increasing attention, as this not only deepens the theoretical understanding of these algorithms but also aligns more closely with practice. Notably, people have demonstrated that one of the most famous, simple, and popular methods, Stochastic Gradient Descent ($\mathtt{SGD}$), provably achieves last-iterate convergence. However, the existing results may have limited applicability, since they typically rely on strong conditions on the gradient noise (e.g., an exponentially decaying tail) rather than the more realistic heavy-tailed noise. Unfortunately, when faced with heavy-tailed noise, $\mathtt{SGD}$ is known to exhibit undesirable behavior or even fail to converge. To address the challenge of heavy-tailed noise, people have proposed $\mathtt{Clipped}\text{-}\mathtt{SGD}$, an algorithm that combines $\mathtt{SGD}$ with a simple mechanism, gradient clipping. Although $\mathtt{Clipped}\text{-}\mathtt{SGD}$ performs well in practice, its last-iterate convergence property remains highly underexplored. In this work, we prove the first optimal last-iterate rates in high probability for $\mathtt{Clipped}\text{-}\mathtt{SGD}$ under heavy-tailed noise, thereby closing a gap in the literature.
Abstract:In stochastic (strongly) convex optimization, establishing last-iterate convergence guarantees for first-order methods has recently garnered increasing attention, as this not only deepens the theoretical understanding of these algorithms but also aligns more closely with practice. Notably, people have demonstrated that one of the most famous, simple, and popular methods, Stochastic Gradient Descent ($\mathtt{SGD}$), provably achieves last-iterate convergence. However, the existing results may have limited applicability, since they typically rely on strong conditions on the gradient noise (e.g., an exponentially decaying tail) rather than the more realistic heavy-tailed noise. Unfortunately, when faced with heavy-tailed noise, $\mathtt{SGD}$ is known to exhibit undesirable behavior or even fail to converge. To address the challenge of heavy-tailed noise, people have proposed $\mathtt{Clipped}\text{-}\mathtt{SGD}$, an algorithm that combines $\mathtt{SGD}$ with a simple mechanism, gradient clipping. Although $\mathtt{Clipped}\text{-}\mathtt{SGD}$ performs well in practice, its last-iterate convergence property remains highly underexplored. In this work, we prove the first optimal last-iterate rates in high probability for $\mathtt{Clipped}\text{-}\mathtt{SGD}$ under heavy-tailed noise, thereby closing a gap in the literature.
Optimization and Control (math.OC); Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2610.21026 [math.OC]
(or arXiv:2610.21026v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2610.21026 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 267 words
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