Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity
Abstract
We consider the convex-concave saddle point problem $\min_{x}\max_{y} f(x)+y^\top A x-g(y)$ where $f$ is smooth and convex and $g$ is smooth and strongly convex. We prove that if the coupling matrix $A$ has full column rank, the vanilla primal-dual gradient method can achieve linear convergence even if $f$ is not strongly convex. Our result generalizes previous work which either requires $f$ and $g$ to be quadratic functions or requires proximal mappings for both $f$ and $g$. We adopt a novel analysis technique that in each iteration uses a "ghost" update as a reference, and show that the iterates in the primal-dual gradient method converge to this "ghost" sequence. Using the same technique we further give an analysis for the primal-dual stochastic variance reduced gradient method for convex-concave saddle point problems with a finite-sum structure.
BibTeX
@InProceedings{pmlr-v89-du19b,
title = {Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity},
author = {Du, Simon S. and Hu, Wei},
booktitle = {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics},
pages = {196--205},
year = {2019},
editor = {Chaudhuri, Kamalika and Sugiyama, Masashi},
volume = {89},
series = {Proceedings of Machine Learning Research},
month = {16--18 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v89/du19b/du19b.pdf},
url = {https://proceedings.mlr.press/v89/du19b.html},
abstract = {We consider the convex-concave saddle point problem $\min_{x}\max_{y} f(x)+y^\top A x-g(y)$ where $f$ is smooth and convex and $g$ is smooth and strongly convex. We prove that if the coupling matrix $A$ has full column rank, the vanilla primal-dual gradient method can achieve linear convergence even if $f$ is not strongly convex. Our result generalizes previous work which either requires $f$ and $g$ to be quadratic functions or requires proximal mappings for both $f$ and $g$. We adopt a novel analysis technique that in each iteration uses a "ghost" update as a reference, and show that the iterates in the primal-dual gradient method converge to this "ghost" sequence. Using the same technique we further give an analysis for the primal-dual stochastic variance reduced gradient method for convex-concave saddle point problems with a finite-sum structure.}
}