On lower iteration complexity bounds for the convex concave saddle point problems

In this paper, we study the lower iteration complexity bounds for finding saddle point of a strongly convex and concave problem: $\min_x\max_yF(x,y)$. We restrict classes algorithms in our investigation to be either pure first-order methods or using proximal mappings. The existing bound result type problems is obtained via framework monotone variational inequality problems, which corresponds case where gradient Lipschitz constants ($L_x, L_y$ $L_{xy}$) strong convexity/concavity ($\mu_x$ $\mu_y$) are uniform with respect variables $x$ $y$. However, specific min-max problem these parameters naturally different. Therefore, one led best possible bounds, models. paper present following results. For class algorithms, $\Omega\left(\sqrt{\frac{L_x}{\mu_x}+\frac{L_{xy}^2}{\mu_x\mu_y}+\frac{L_y}{\mu_y}}\cdot\ln\left(\frac{1}{\epsilon}\right)\right)$, term $\frac{L_{xy}^2}{\mu_x\mu_y}$ explains how coupling influences complexity. Under several special parameter regimes, has been achieved by corresponding optimal algorithms. whether not under general regime remains open. Additionally, bilinear given availability certain operators, $\Omega\left(\sqrt{\frac{L_{xy}^2}{\mu_x\mu_y}+1}\cdot\ln(\frac{1}{\epsilon})\right)$ established have already developed literature.