Two steps at a time -- taking GAN training in stride with Tseng's method

Motivated by the training of Generative Adversarial Networks (GANs), we study methods for solving minimax problems with additional nonsmooth regularizers. We do so by employing \emph{monotone operator} theory, in particular the \emph{Forward-Backward-Forward (FBF)} method, which avoids the known issue of limit cycling by correcting each update by a second gradient evaluation. Furthermore, we propose a seemingly new scheme which recycles old gradients to mitigate the additional computational cost. In doing so we rediscover a known method, related to \emph{Optimistic Gradient Descent Ascent (OGDA)}. For both schemes we prove novel convergence rates for convex-concave minimax problems via a unifying approach. The derived error bounds are in terms of the gap function for the ergodic iterates. For the deterministic and the stochastic problem we show a convergence rate of $\mathcal{O}(1/k)$ and $\mathcal{O}(1/\sqrt{k})$, respectively. We complement our theoretical results with empirical improvements in the training of Wasserstein GANs on the CIFAR10 dataset.