Last Iterate Is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems
Abstract
In this paper we study the smooth convex-concave saddle point problem. Specifically, we analyze the last iterate convergence properties of the Extragradient (EG) algorithm. It is well known that the ergodic (averaged) iterates of EG converge at a rate of $O(1/T)$ (Nemirovski, 2004). In this paper, we show that the last iterate of EG converges at a rate of $O(1/\sqrt{T})$. To the best of our knowledge, this is the first paper to provide a convergence rate guarantee for the last iterate of EG for the smooth convex-concave saddle point problem. Moreover, we show that this rate is tight by proving a lower bound of $\Omega(1/\sqrt{T})$ for the last iterate. This lower bound therefore shows a quadratic separation of the convergence rates of ergodic and last iterates in smooth convex-concave saddle point problems.
Cite
Text
Golowich et al. "Last Iterate Is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems." Conference on Learning Theory, 2020.Markdown
[Golowich et al. "Last Iterate Is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems." Conference on Learning Theory, 2020.](https://mlanthology.org/colt/2020/golowich2020colt-last/)BibTeX
@inproceedings{golowich2020colt-last,
title = {{Last Iterate Is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems}},
author = {Golowich, Noah and Pattathil, Sarath and Daskalakis, Constantinos and Ozdaglar, Asuman},
booktitle = {Conference on Learning Theory},
year = {2020},
pages = {1758-1784},
volume = {125},
url = {https://mlanthology.org/colt/2020/golowich2020colt-last/}
}