On the Last-Iterate Convergence of Clipped Gradient Methods

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.