Last-Iterate Convergence of Clipped SGD with Linear Decay

arXiv:2610.18555v1 Announce Type: new 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<ple2. For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order O(LR^2log(1/elta)/T+sigma R(log(1/elta)/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/elta) replaced by log(1/elta)+q. The corresponding lower bounds for the clean and corrupted oracle models are also provided.

Source: arXiv cs.LG | 2026-10-14