  We generalize gradient descent with momentum for optimization in
differentiable games to have complex-valued momentum. We give theoretical
motivation for our method by proving convergence on bilinear zero-sum games for
simultaneous and alternating updates. Our method gives real-valued parameter
updates, making it a drop-in replacement for standard optimizers. We
empirically demonstrate that complex-valued momentum can improve convergence in
realistic adversarial games - like generative adversarial networks - by showing
we can find better solutions with an almost identical computational cost. We
also show a practical generalization to a complex-valued Adam variant, which we
use to train BigGAN to better inception scores on CIFAR-10.
