The empirical success of Generative Adversarial Networks (GANs) caused an increasing interest in theoretical research. The statistical literature is mainly focused on Wasserstein GANs and generalizations thereof, which especially allow for good dimension reduction properties. Statistical results for Vanilla GANs, the original optimization problem, are still rather limited and require assumptions such as smooth activation functions and equal dimensions of the latent space and the ambient space. To bridge this gap, we draw a connection from Vanilla GANs to the Wasserstein distance. By doing so, existing results for Wasserstein GANs can be extended to Vanilla GANs. In particular, we obtain an oracle inequality for Vanilla GANs in Wasserstein distance. The assumptions of this oracle inequality are designed to be satisfied by network architectures commonly used in practice, such as feedforward ReLU networks. By providing a quantitative result for the approximation of a Lipschitz function by a feedforward ReLU network with bounded H\"older norm, we conclude a rate of convergence for Vanilla GANs as well as Wasserstein GANs as estimators of the unknown probability distribution.
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