Can SGD Handle Heavy-Tailed Noise?
This work addresses a foundational theoretical gap in optimization for machine learning practitioners, showing that vanilla SGD is robust under heavy-tailed noise without needing modifications.
The paper tackles the problem of whether vanilla Stochastic Gradient Descent (SGD) can handle heavy-tailed noise, common in machine learning, by establishing sharp convergence guarantees under minimal assumptions. It shows that SGD achieves minimax optimal sample complexities, such as O(ε^(-p/(p-1))) for convex problems, proving it remains effective even with unbounded variance.
Stochastic Gradient Descent (SGD) is a cornerstone of large-scale optimization, yet its theoretical behavior under heavy-tailed noise -- common in modern machine learning and reinforcement learning -- remains poorly understood. In this work, we rigorously investigate whether vanilla SGD, devoid of any adaptive modifications, can provably succeed under such adverse stochastic conditions. Assuming only that stochastic gradients have bounded $p$-th moments for some $p \in (1, 2]$, we establish sharp convergence guarantees for (projected) SGD across convex, strongly convex, and non-convex problem classes. In particular, we show that SGD achieves minimax optimal sample complexity under minimal assumptions in the convex and strongly convex regimes: $\mathcal{O}(\varepsilon^{-\frac{p}{p-1}})$ and $\mathcal{O}(\varepsilon^{-\frac{p}{2(p-1)}})$, respectively. For non-convex objectives under Hölder smoothness, we prove convergence to a stationary point with rate $\mathcal{O}(\varepsilon^{-\frac{2p}{p-1}})$, and complement this with a matching lower bound specific to SGD with arbitrary polynomial step-size schedules. Finally, we consider non-convex Mini-batch SGD under standard smoothness and bounded central moment assumptions, and show that it also achieves a comparable $\mathcal{O}(\varepsilon^{-\frac{2p}{p-1}})$ sample complexity with a potential improvement in the smoothness constant. These results challenge the prevailing view that heavy-tailed noise renders SGD ineffective, and establish vanilla SGD as a robust and theoretically principled baseline -- even in regimes where the variance is unbounded.