Stochastic Recursive Gradient Descent Ascent for Stochastic Nonconvex-Strongly-Concave Minimax Problems

We consider nonconvex-concave minimax optimization problems of the form min(x) max(y is an element of Y) f (x; y), where f is strongly-concave in y but possibly nonconvex in x and Y is a convex and compact set. We focus on the stochastic setting, where we can only access an unbiased stochastic gradient estimate of f at each iteration. This formulation includes many machine learning applications as special cases such as robust optimization and adversary training. We are interested in finding an O(epsilon)-stationary point of the function Phi(center dot) = max(y is an element of Y) f (center dot, y). The most popular algorithm to solve this problem is stochastic gradient decent ascent, which requires O(kappa 3 epsilon(-4)) stochastic gradient evaluations, where kappa is the condition number. In this paper, we propose a novel method called Stochastic Recursive gradiEnt Descent Ascent (SREDA), which estimates gradients more efficiently using variance reduction. This method achieves the best known stochastic gradient complexity of O(kappa 3 epsilon(-4)), and its dependency on epsilon is optimal for this problem.