In this article, we propose a novel loss function for training generative adversarial networks (GANs) aiming toward deeper theoretical understanding as well as improved stability and performance for the underlying optimization problem. The new loss function is based on cumulant generating functions (CGFs) giving rise to Cumulant GAN. Relying on a recently derived variational formula, we show that the corresponding optimization problem is equivalent to Rényi divergence minimization, thus offering a (partially) unified perspective of GAN losses: the Rényi family encompasses Kullback-Leibler divergence (KLD), reverse KLD, Hellinger distance, and χ2 -divergence. Wasserstein GAN is also a member of cumulant GAN. In terms of stability, we rigorously prove the linear convergence of cumulant GAN to the Nash equilibrium for a linear discriminator, Gaussian distributions, and the standard gradient descent ascent algorithm. Finally, we experimentally demonstrate that image generation is more robust relative to Wasserstein GAN and it is substantially improved in terms of both inception score (IS) and Fréchet inception distance (FID) when both weaker and stronger discriminators are considered.