Lower Complexity Bounds for Finite-Sum Convex-Concave Minimax Optimization Problems

This paper studies the lower bound complexity for minimax optimization problem whose objective function is the average of n individual smooth convex-concave functions. We consider the algorithm which has access to gradient and proximal oracle for each individual component. For the strongly-convex-strongly-concave case, we prove such an algorithm can not reach an ε-saddle point in fewer than Ω ((n + κ) log(1/ε)) iterations, where κ is the condition number of the objective function. This lower bound matches the upper bound of the existing proximal incremental first-order oracle algorithm in some specific case. We develop a novel construction to show the above result, which partitions the tridiagonal matrix of classical examples into n groups. This construction is friendly to the analysis of incremental gradient and proximal oracle and we also extend the analysis to general convex-concave cases.