Vanilla SGD with Momentum Survives Heavy-Tailed Noise: Convergence Analysis without Gradient Clipping or Normalization

Many stochastic gradient methods are believed not to converge when the noise in stochastic gradients has only a finite $p$-th moment for $p\in\left(1,2\right)$, a setting known as the heavy-tailed noise assumption. However, some recent studies have found that Stochastic Gradient Descent ($\textsf{SGD}$), without any modification to its update rule, can surprisingly converge in expectation for convex problems with bounded domains, highlighting the potential of classical stochastic gradient methods. Inspired by this recent progress, we provide a comprehensive study of stochastic optimization under heavy-tailed noise and establish new in-expectation convergence results for Stochastic Mirror Descent ($\textsf{SMD}$) and Accelerated Stochastic Mirror Descent ($\textsf{ASMD}$) in convex optimization, and for $\textsf{SGD}$ and Stochastic Gradient Descent with Momentum ($\textsf{SGDM}$) in nonconvex optimization. Notably, our results not only hold without algorithmic changes but also avoid restrictive assumptions, such as bounded domains, imposed in prior work. More importantly, our analysis provides a new, elegant, and powerful framework for studying heavy-tailed stochastic optimization, opening a new route to understanding first-order stochastic gradient methods.

Classical convergence guarantees for stochastic gradient methods typically assume Lipschitz-smooth objectives and finite-variance gradient noise, both frequently violated in practice. In contrast, we study nonconvex stochastic optimization under the joint relaxation of these assumptions: objectives with $(L,s)$-H"older continuous gradients, $s\in(0,1]$, and gradient noise satisfying only a bounded $\alpha$-th moment condition for $\alpha\in(1,2]$. We establish three convergence results. Firstly, that standard SGD converges at rate $O(T^{-s/(1+s)})$ whenever $\alpha\ge1+s$, extending the classical nonconvex SGD rate to heavy-tailed noise and H"older smoothness simultaneously. Secondly, we analyze $\delta$-regularized gradient clipping ($\delta$-GClip), a provable trainer of wide and deep nets, and establish a stationarity rate of $O(T^{-2s(\alpha-1)/[(1+s)(2\alpha-1)]})$ under the same condition. Thirdly, we analyze standard gradient clipping (G-Clip) and show that it recovers the above rate for $\alpha\ge1+s$ while in the very heavy-tailed regime $\alpha<1+s$, it has a convergence rate $O(T^{-2s(\alpha-1)/[(\alpha-1)+s(2\alpha-1)]})$ --- the first convergence guarantee in this regime for any stochastic gradient based method.

Many tasks in modern machine learning are observed to involve heavy-tailed gradient noise during the optimization process. To manage this realistic and challenging setting, new mechanisms, such as gradient clipping and gradient normalization, have been introduced to ensure the convergence of first-order algorithms. However, adaptive gradient methods, a famous class of modern optimizers that includes popular $\mathtt{Adam}$ and $\mathtt{AdamW}$, often perform well even without any extra operations mentioned above. It is therefore natural to ask whether adaptive gradient methods can converge under heavy-tailed noise without any algorithmic changes. In this work, we take the first step toward answering this question by investigating a special case, $\mathtt{AdaGrad}$, the origin of adaptive gradient methods. We provide the first provable convergence rate for $\mathtt{AdaGrad}$ in non-convex optimization when the tail index $p$ satisfies $4/3<p\leq2$. Notably, this result is achieved without requiring any prior knowledge of $p$ and is hence adaptive to the tail index. In addition, we develop an algorithm-dependent lower bound, suggesting that the existing minimax rate for heavy-tailed optimization is not attainable by $\mathtt{AdaGrad}$. Lastly, we consider $\mathtt{AdaGrad}\text{-}\mathtt{Norm}$, a popular variant of $\mathtt{AdaGrad}$ in theoretical studies, and show an improved rate that holds for any $1<p\leq2$ under an extra mild assumption.