Gradient Clipping Improves AdaGrad when the Noise Is Heavy-Tailed

Saveliy Chezhegov, Yaroslav Klyukin, Andrei Semenov, Aleksandr Beznosikov, Alexander V. Gasnikov, Samuel Horváth, Martin Takác, Eduard Gorbunov

Abstract: Methods with adaptive stepsizes, such as AdaGrad and Adam, are essential for training modern Deep Learning models, especially Large Language Models. Typically, the noise in the stochastic gradients is heavy-tailed for the later ones. Gradient clipping provably helps to achieve good high-probability convergence for such noises. However, despite the similarity between AdaGrad/Adam and Clip-SGD, the current understanding of the high-probability convergence of AdaGrad/Adam-type methods is limited in this case. In this work, we prove that AdaGrad/Adam (and their delayed version) can have provably bad high-probability convergence if the noise is heavy-tailed. We also show that gradient clipping fixes this issue, i.e., we derive new high-probability convergence bounds with polylogarithmic dependence on the confidence level for AdaGrad-Norm and Adam-Norm with clipping and with/without delay for smooth convex/non-convex stochastic optimization with heavy-tailed noise. Our empirical evaluations highlight the superiority of clipped versions of AdaGrad/Adam-Norm in handling the heavy-tailed noise.