Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity

Simon S. Du, Wei Hu
Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, PMLR 89:196-205, 2019.

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.

Cite this Paper

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.} }
Endnote
%0 Conference Paper %T Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity %A Simon S. Du %A Wei Hu %B Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Masashi Sugiyama %F pmlr-v89-du19b %I PMLR %P 196--205 %U https://proceedings.mlr.press/v89/du19b.html %V 89 %X 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.
APA
Du, S.S. & Hu, W.. (2019). Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity. Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 89:196-205 Available from https://proceedings.mlr.press/v89/du19b.html.

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