Gradient Normalization Provably Benefits Nonconvex SGD under Heavy-Tailed Noise

This paper investigates the roles of gradient normalization and clipping in ensuring the convergence of Stochastic Gradient Descent (SGD) under heavy-tailed noise. While existing approaches consider gradient clipping indispensable for SGD convergence, we theoretically demonstrate that gradient normalization alone without clipping is sufficient to ensure convergence. Furthermore, we establish that combining gradient normalization with clipping offers significantly improved convergence rates compared to using either technique in isolation, particularly as gradient noise diminishes. With these results, our work provides the first theoretical evidence demonstrating the benefits of gradient normalization in SGD under heavy-tailed noise. Finally, we introduce an accelerated SGD variant that incorporates both gradient normalization and clipping, further enhancing convergence rates under heavy-tailed noise.