• Corpus ID: 55613424

A Primal-Dual Algorithm for General Convex-Concave Saddle Point Problems

@article{Hamedani2018APA,
  title={A Primal-Dual Algorithm for General Convex-Concave Saddle Point Problems},
  author={Erfan Yazdandoost Hamedani and Necdet Serhat Aybat},
  journal={arXiv: Optimization and Control},
  year={2018}
}

• E. Y. Hamedani, N. S. Aybat
• Published 4 March 2018
• Mathematics, Computer Science
• arXiv: Optimization and Control

In this paper we propose a primal-dual algorithm with a momentum term that can be viewed as a generalization of the method proposed by Chambolle and Pock in 2016 to solve saddle point problems defined by a convex-concave function $\mathcal{L}(x,y)=f(x)+\Phi(x,y)-h(y)$ with a general coupling term $\Phi(x,y)$ that is not assumed to be bilinear. Given a saddle point $(x^*,y^*)$, assuming $\nabla_y\Phi(\cdot,\cdot)$ is Lipschitz and $\nabla_x\Phi(\cdot,y)$ is Lipschitz in $x$ for any fixed $y$, we derive error bounds in terms of $\mathcal{L}(\bar{x}_k,x^*)-\mathcal{L}(y^*,\bar{y}_k)$ for the ergodic sequence $\{\bar{x}_k,\bar{y}_k\}$; in particular, we show $\mathcal{O}(1/k)$ rate that when the problem is merely convex in $x$. Furthermore, assuming $\Phi(x,\cdot)$ is linear in $y$ for each fixed $x$ and $f$ is strongly convex, we can obtain the ergodic convergence rate of $\mathcal{O}(1/k^2)$ - we are not aware of any other work in the related literature showing $\mathcal{O}(1/k^2)$ rate when $\Phi$ is not bilinear. We tested our method for solving kernel matrix learning problem, and compare it against the Mirror-prox algorithm and interior point methods.