An Improved Analysis of the Last Iterate of the SubGradient Method in the Plane

Guglielmo Beretta; Tommaso Cesari; Roberto Colomboni; Andrea Paudice

An Improved Analysis of the Last Iterate of the SubGradient Method in the Plane

Optimization and Control 2026-07-17 v1

Authors: Guglielmo Beretta , Tommaso Cesari , Roberto Colomboni , Andrea Paudice

Abstract

We study the last iterate of the subGradient Method (sGM) for convex Lipschitz objectives defined on $\mathbb R^2$ . We prove that for a finite horizon $n$ and a constant stepsize $\eta = \Theta(1/\sqrt{n})$ , the last iterate achieves an optimization error of order $1/\sqrt{n}$ , showing that the extra $\log{n}$ factor appearing in high dimensions is unnecessary in the plane. This makes progress on an open problem posed by Koren and Segal in 2020 on the optimal error of the last iterate of sGM in fixed dimension.

Keywords

subgradient method last iterate convex optimization Lipschitz functions optimization error fixed dimension

Cite

@article{arxiv.2607.15980,
  title  = {An Improved Analysis of the Last Iterate of the SubGradient Method in the Plane},
  author = {Guglielmo Beretta and Tommaso Cesari and Roberto Colomboni and Andrea Paudice},
  journal= {arXiv preprint arXiv:2607.15980},
  year   = {2026}
}