Last-Iterate Convergence of Clipped SGD with Linear Decay

Abstract

In this paper, we study the last-iterate convergence of clipped stochastic gradient descent under heavy-tailed noise. We assume that the stochastic gradients are unbiased and that the noise has a bounded central $p$-th moment, where $1<p\le 2$. For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order $O(L{R}^2\log (1/\delta )/T+\sigma R(\log (1/\delta )/T{)}^{(p-1)/p})$. We further establish last-iterate guarantees for nonsmooth composite objectives and smooth composite objectives with proximal updates. In addition, when a causal adversary replaces at most $q$ gradients, the same clipped update achieves a bound with $\log (1/\delta )$ replaced by $\log (1/\delta )+q$. The corresponding lower bounds for the clean and corrupted oracle models are also provided.