  Despite the success of generative adversarial networks (GANs) in generating
visually appealing images, they are notoriously challenging to train. In order
to stabilize the learning dynamics in minimax games, we propose a novel
recursive reasoning algorithm: Level $k$ Gradient Play (Lv.$k$ GP) algorithm.
In contrast to many existing algorithms, our algorithm does not require
sophisticated heuristics or curvature information. We show that as $k$
increases, Lv.$k$ GP converges asymptotically towards an accurate estimation of
players' future strategy. Moreover, we justify that Lv.$\infty$ GP naturally
generalizes a line of provably convergent game dynamics which rely on
predictive updates. Furthermore, we provide its local convergence property in
nonconvex-nonconcave zero-sum games and global convergence in bilinear and
quadratic games. By combining Lv.$k$ GP with Adam optimizer, our algorithm
shows a clear advantage in terms of performance and computational overhead
compared to other methods. Using a single Nvidia RTX3090 GPU and 30 times fewer
parameters than BigGAN on CIFAR-10, we achieve an FID of 10.17 for
unconditional image generation within 30 hours, allowing GAN training on common
computational resources to reach state-of-the-art performance.
