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arXiv · 2610.18555

Last-Iterate Convergence of Clipped SGD with Linear Decay

Wei Jiang·Haomin Bai·Zechao Li
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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 $1<p\le2$. For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order $O(LR^2\log(1/δ)/T+σR(\log(1/δ)/T)^{(p-1)/p})$. 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.

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Wei Jiang, Haomin Bai, Zechao Li. 2026-10-13. Last-Iterate Convergence of Clipped SGD with Linear Decay. https://arxiv.org/abs/2610.18555

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