Stronger constraints for smooth min-max games

Abstract

Saddle point problems with smooth convex-concave objective functions are often used to model min-max problems arising in machine learning. First-order methods are the standard paradigm for solving such problems. Therefore, it is important to know how those methods behave in the worst-case scenarios. In order to derive the guarantees, one would require the inequalities that appropriately constrain the iterates, gradients and function values. In this paper, we present stronger constraints for smooth convex-concave functions and show that they could allow tighter upper bounds for first-order methods.