Clipped SGD method improves convergence under heavy-tailed noise
On the Last-Iterate Convergence of Clipped Gradient Methods
Machine Learning
Summary
When computers learn by adjusting step-by-step, sometimes the updates are noisy and unpredictable, causing problems. Traditional methods work well if the noise behaves nicely, but not when it has wild variations called heavy tails. The authors studied a technique called Clipped SGD that cuts down the impact of extreme noisy updates. They proved mathematically that this approach reliably converges even with heavy-tailed noise, solving a known problem in machine learning optimization.
What this means in practice
- •For machine learning engineers: Design training algorithms that remain stable and reliable when data noise is heavy-tailed by using clipped gradients with guaranteed convergence.
- •For signal processing teams: Develop optimization routines that converge reliably under unpredictable noise patterns common in real-world signal data using gradient clipping methods.
A theory result. No direct application yet.
Authors
Zijian Liu
Abstract
In stochastic (strongly) convex optimization, establishing last-iterate convergence guarantees for first-order methods has recently garnered increasing attention, as this not only deepens the theoretical understanding of these algorithms but also aligns more closely with practice. Notably, people have demonstrated that one of the most famous, simple, and popular methods, Stochastic Gradient Descent ($\mathtt{SGD}$), provably achieves last-iterate convergence. However, the existing results may have limited applicability, since they typically rely on strong conditions on the gradient noise (e.g., an exponentially decaying tail) rather than the more realistic heavy-tailed noise. Unfortunately, when faced with heavy-tailed noise, $\mathtt{SGD}$ is known to exhibit undesirable behavior or even fail to converge. To address the challenge of heavy-tailed noise, people have proposed $\mathtt{Clipped}\text{-}\mathtt{SGD}$, an algorithm that combines $\mathtt{SGD}$ with a simple mechanism, gradient clipping. Although $\mathtt{Clipped}\text{-}\mathtt{SGD}$ performs well in practice, its last-iterate convergence property remains highly underexplored. In this work, we prove the first optimal last-iterate rates in high probability for $\mathtt{Clipped}\text{-}\mathtt{SGD}$ under heavy-tailed noise, thereby closing a gap in the literature.