We study the problem of alleviating the instability issue in the GAN training procedure via new architecture design. The discrepancy between the minimax and maximin objective values could serve as a proxy for the difficulties that the alternating gradient descent encounters in the optimization of GANs. In this work, we give new results on the benefits of multi-generator architecture of GANs. We show that the minimax gap shrinks to $epsilon$ as the number of generators increases with rate $widetilde{O}(1/epsilon)$. This improves over the best-known result of $widetilde{O}(1/epsilon^2)$. At the core of our techniques is a novel application of Shapley-Folkman lemma to the generic minimax problem, where in the literature the technique was only known to work when the objective function is restricted to the Lagrangian function of a constraint optimization problem. Our proposed Stackelberg GAN performs well experimentally in both synthetic and real-world datasets, improving Frechet Inception Distance by $14.61%$ over the previous multi-generator GANs on the benchmark datasets.
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