Stochastic gradient methods converge despite rough data and noise
Convergence of Stochastic Gradient Methods under Heavy-Tailed Noise and Hölder Smoothness
Machine Learning
Summary
Stochastic gradient methods help machines learn by making small, random updates based on data. These methods usually assume smoothness in the problem and limited randomness (noise) in updates. The authors show that even when these assumptions are weakened—such as when the problem is less smooth and noise is heavy-tailed—these methods still reliably find good solutions under new mathematical conditions. They provide specific rates of how fast the methods converge, improving understanding of training deep neural networks with noisy data.
What this means in practice
- •For machine learning engineers: Design and run training algorithms to better handle noisy, irregular data using gradient clipping informed by the new convergence guarantees.
- •For ai infrastructure teams: Improve system robustness in large-scale neural network training by choosing stochastic gradient methods that maintain convergence despite heavy noise.
A theory result. No direct application yet.
Authors
Misbah Uz Zaman, Anirbit Mukherjee
Abstract
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 $(L,s)$-Hölder continuous gradients, $s\in(0,1]$, and gradient noise satisfying only a bounded $α$-th moment condition for $α\in(1,2]$. We establish three convergence results. Firstly, that standard SGD converges at rate $O(T^{-s/(1+s)})$ whenever $α\ge1+s$, extending the classical nonconvex SGD rate to heavy-tailed noise and Hölder smoothness simultaneously. Secondly, we analyze $δ$-regularized gradient clipping ($δ$-GClip), a provable trainer of wide and deep nets, and establish a stationarity rate of $O(T^{-2s(α-1)/[(1+s)(2α-1)]})$ under the same condition. Thirdly, we analyze standard gradient clipping (G-Clip) and show that it recovers the above rate for $α\ge1+s$ while in the very heavy-tailed regime $α<1+s$, it has a convergence rate $O(T^{-2s(α-1)/[(α-1)+s(2α-1)]})$ --- the first convergence guarantee in this regime for any stochastic gradient based method.