Near Optimal Stochastic Algorithms for Finite-Sum Unbalanced Convex-Concave Minimax Optimization

This paper considers stochastic first-order algorithms for convex-concave minimax problems of the form $\min{\bf x}\max{\bf y}f(\bf x, \bf y)$, where $f$ can be presented by the average of $n$ individual components which are $L$-average smooth. For $\mux$-strongly-convex-$\muy$-strongly-concave setting, we propose a new method which could find a $\varepsilon$-saddle point of the problem in $\tilde{\mathcal O} \big(\sqrt{n(\sqrt{n}+\kappax)(\sqrt{n}+\kappay)}\log(1/\varepsilon)\big)$ stochastic first-order complexity, where $\kappax\triangleq L/\mux$ and $\kappay\triangleq L/\muy$. This upper bound is near optimal with respect to $\varepsilon$, $n$, $\kappax$ and $\kappay$ simultaneously. In addition, the algorithm is easily implemented and works well in practical. Our methods can be extended to solve more general unbalanced convex-concave minimax problems and the corresponding upper complexity bounds are also near optimal.