A PrimalDual Algorithm with Line Search for General ConvexConcave Saddle Point Problems
Abstract
In this paper, we propose a primaldual algorithm with a novel momentum term using the partial gradients of the coupling function 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 convexconcave 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. Assuming $\nabla_x\Phi(\cdot,y)$ is Lipschitz for any fixed $y$, and $\nabla_y\Phi(\cdot,\cdot)$ is Lipschitz, we show that the iterate sequence converges to a saddle point; and for any $(x,y)$, we derive error bounds in terms of $\mathcal L(\bar{x}_k,y)\mathcal L(x,\bar{y}_k)$ for the ergodic sequence $\{\bar{x}_k,\bar{y}_k\}$. In particular, we show $\mathcal O(1/k)$ rate when the problem is merely convex in $x$. Furthermore, assuming $\Phi(x,\cdot)$ is linear for each fixed $x$ and $f$ is strongly convex, we obtain the ergodic convergence rate of $\mathcal O(1/k^2)$  we are not aware of another singleloop method in the related literature achieving the same rate when $\Phi$ is not bilinear. Finally, we propose a backtracking technique which does not require the knowledge of Lipschitz constants while ensuring the same convergence results. We also consider convex optimization problems with nonlinear functional constraints and we show that using the backtracking scheme, the optimal convergence rate can be achieved even when the dual domain is unbounded. We tested our method against other stateoftheart firstorder algorithms and interiorpoint methods for solving quadratically constrained quadratic problems with synthetic data, the kernel matrix learning, and regression with fairness constraints arising in machine learning.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 DOI:
 10.48550/arXiv.1803.01401
 arXiv:
 arXiv:1803.01401
 Bibcode:
 2018arXiv180301401Y
 Keywords:

 Mathematics  Optimization and Control
 EPrint:
 linesearch is added