Stochastic gradient methods with clipping and noise converge reliably
Almost Sure Convergence Analysis of Stochastic Gradient Methods with Clipping and Additive Noise
Machine Learning
Summary
Training machine learning models often uses a method called stochastic gradient descent, where small steps improve the model based on random samples. Adding clipping (cutting off very large steps) and noise can help make training more robust and private but complicates how the learning progresses. The authors show that despite these complications, the method still converges to good solutions under reasonable conditions. They also demonstrate that similar convergence holds for related methods that add momentum, helping us understand the stability of these popular training techniques.
What this means in practice
- •For machine learning engineers: Guarantee stable training when using clipped gradients and noise for robustness or privacy in model development.
- •For data privacy teams: Ensure reliable optimization when adding noise to gradients to protect data privacy during model training.
A theory result. No direct application yet.
Authors
Amartya Mukherjee, Jun Liu
Abstract
Stochastic gradient descent (SGD) with gradient clipping and additive noise has become a standard technique for training machine learning models, particularly in applications requiring robustness or privacy guarantees. However, clipping introduces a bias in stochastic gradients, while additive noise introduces additional variance, making the long-run behaviour of individual optimization trajectories difficult to characterize. In this work, we prove that SGD with clipping and additive Gaussian noise (SGD-CN) converges almost surely (a.s.) under smoothness and uniformly bounded stochastic-gradient noise assumptions, provided the step sizes satisfy some standard decaying conditions. Our analysis extends to momentum variants such as the stochastic heavy ball and Nesterov's accelerated gradient, where we show that careful energy constructions yield similar guarantees. These results provide stronger theoretical foundations for understanding the pathwise behaviour of clipped stochastic gradient methods and suggest that, despite the bias and noise introduced by clipping and perturbation, the algorithm remains stable in both convex and nonconvex regimes.