is obtained as a natural extension of gradient descent to the competitive setting. Recall that in the single player setting, the gradient descent update is obtained as the optimal solution to a regularized linear approximation of the cost function. In the same spirit, the update of CGD is given by the Nash equilibrium of a regularized *bilinear* approximation of the underlying game. The use of a bilinear– as opposed to linear approximation lets the local approximation preserve the competitive nature of the problem, significantly improving stability. We prove (local) convergence results of this algorithm in the case of (locally) convex-concave zero-sum games. We also show that stronger interactions between the two players only improve convergence, without requiring an adaptation of the stepsize. In comparison, the existing methods need to reduce the stepsize to match the increase of the interactions to avoid divergence, which we illustrate on a series of polynomial test cases considered in previous works.\
We begin our numerical experiments by trying to use a GAN on a bimodal Gaussian mixture model. Even in this simple example, trying five different (constant) stepsizes under RMSProp, the existing methods diverge. The typical solution would be to decay the learning rate. However 