Alternating Proximal-Gradient Steps for (Stochastic) Nonconvex-Concave Minimax Problems

.Minimax problems of the form \(min_x max_y Ψ (x,y)\) have attracted increased interest largely due to advances in machine learning, in particular generative adversarial networks and adversarial learning. These are typically trained using variants of stochastic gradient descent for the two players. Although convex-concave problems are well understood with many efficient solution methods to choose from, theoretical guarantees outside of this setting are sometimes lacking even for the simplest algorithms. In particular, this is the case for alternating gradient descent ascent, where the two agents take turns updating their strategies. To partially close this gap in the literature we prove a novel global convergence rate for the stochastic version of this method for finding a critical point of \(ψ (· ) := max_y Ψ (·,y)\) in a setting which is not convex-concave.Keywordsminimaxsaddle pointnonconvex-concavecomplexityprox-gradient methodstochastic gradient descentMSC codes90C4790C1590C25