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?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(e)-stationary point of the function F(·) = max_y?Y f(·, y). The most popular algorithm to solve this problem is stochastic gradient decent ascent, which requires O(?³e^-4) stochastic gradient evaluations, where ? 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(?³e^-3), and its dependency on e is optimal for this problem.