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Weak-$L^α$ Noise, Strong Guarantees: Gradient Clipping under Affine Heavy Tails

Alyona Nekrasova, Makar Sidorov, Savelii Chezhegov, Alexander Vladimirovich Gasnikov, Margarita M. Preobrazhenskaia, Igor Preobrazhensky, Eduard Gorbunov · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28107 · cited by 0 works (OpenAlex) · open-access PDF

French

Le clipping fixe de gradient inférieur est un outil clé pour obtenir des garanties de convergence de haute probabilité pour les méthodes stochastiques de premier ordre sous un bruit de queue lourde. Les analyses existantes supposent généralement un moment central limité de l'ordre $α\in(1,2)$, ce qui exclut les distributions avec des queues de l'ordre $u^{-α}$ et sa variante accélérée Clip-SSTM à l'exponent critique. Dans ce document, nous considérons une optimisation convexe lisse sous un bruit de queue lourde affine, dont l'échelle peut croître avec la norme de gradient. Nous dérivons des garanties de convergence de haute probabilité pour le SGD (Clip-SGD)

Machine translation (Meta M2M100); the English original follows.

Abstract

Gradient clipping is a key tool for obtaining high-probability convergence guarantees for stochastic first-order methods under heavy-tailed noise. Existing analyses typically assume a bounded central moment of order $α\in(1,2]$, which excludes distributions with tails of order $u^{-α}$ at the critical exponent. In this paper, we consider smooth convex optimization under affine heavy-tailed noise, whose scale may grow with the gradient norm. We derive high-probability convergence guarantees for clipped SGD (Clip-SGD) and its accelerated variant Clip-SSTM under a weak-$L^α$ tail condition, which is strictly weaker than the corresponding moment bound. For $1<α<2$, the clipping bounds have the same powers of the clipping level as in the moment case, and both methods attain the additive stochastic rate $K^{-(α-1)/α}$. The gradient-dependent noise contributes a faster-decaying term of order $K^{-2(α-1)/α}$, up to logarithmic factors. At $α=2$, we characterize the additional logarithmic factor on the Lorentz scale and establish its necessity for the additive error at fixed confidence. We also analyze Clip-SGD with prescribed clipping levels and derive finite-horizon guarantees for reaching a neighborhood determined by the clipping bias. For additive noise, matching lower bounds show that this neighborhood is unavoidable, up to constants and the natural error cap. Numerical experiments with Pareto noise illustrate the theoretical results.

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Topics: Mathematics, Clipping (morphology), Moment (physics), Applied mathematics, Logarithm, Bounded function

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