Last-Iterate Convergence of Clipped SGD with Linear Decay

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≤2. For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order O(LR2log(1/δ)/T+σ R(log(1/δ)/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/δ) replaced by log(1/δ)+q. The corresponding lower bounds for the clean and corrupted oracle models are also provided.