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≤2. For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order O(LR2log(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.
Related
- On the Last-Iterate Convergence of Clipped Gradient Methods
- Last-Iterate Convergence of Stochastic Gradient Methods for Unbounded Problems
- Large-Step Last-Iterate Convergence of Random-Reshuffling Stochastic Gradient Descent
- Last-Iterate Convergence of Frank--Wolfe for Nonlinear Variational Inequalities on Polytopes
- Last-Iterate Convergence Rate of Normalized Gradient Descent under Hölder Smoothness
BibTeX
@misc{jiang2026lastiterate,
title = {{Last-Iterate Convergence of Clipped SGD with Linear Decay}},
author = {Wei Jiang and Haomin Bai and Zechao Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18555},
eprint = {2610.18555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18555},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Last-Iterate Convergence of Clipped SGD with Linear Decay
AU - Wei Jiang
AU - Haomin Bai
AU - Zechao Li
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18555
UR - https://arxiv.org/abs/2610.18555
AB - 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(LR^2log(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.
ER -