We study the effect of the stochastic gradient noise on the training of generative adversarial networks (GANs) and show that it can prevent the convergence of standard game optimization methods, while the batch version converges. We address this issue with a novel stochastic variance-reduced extragradient (SVRE) optimization algorithm, which for a large class of games improves upon the previous convergence rates proposed in the literature. We observe empirically that SVRE performs similarly to a batch method on MNIST while being computationally cheaper, and that SVRE yields more stable GAN training on standard datasets.