A Parameter-free Algorithm for Convex-concave Min-max Problems.

Parameter-free optimization algorithms refer to algorithms whose convergence rate is optimal with respect to the initial point without any learning rate to tune. They are proposed and well-studied in the online convex optimization literature. However, all the existing parameter-free algorithms can only be used for convex minimization problems. It remains unclear how to design a parameter-free algorithm for convex-concave min-max problems. In fact, the best known convergence rates of the algorithms for solving these problems depend on the size of the domain, rather than on the distance between initial point and the optimal solution. In this paper, we provide the first parameter-free algorithm for several classes of convex-concave problems and establish corresponding state-of-the-art convergence rates, including strictly-convex-strictly-concave min-max problems and min-max problems with non-Euclidean geometry. As a by-product, we utilize the parameter-free algorithm as a subroutine to design a new algorithm, which obtains fast rates for min-max problems with a growth condition. Extensive experiments are conducted to verify our theoretical findings and demonstrate the effectiveness of the proposed algorithm.