the error rate of the discriminative network (i.e., "fool" the discriminator network by producing novel candidates that the discriminator thinks are not synthesized (are part of the true data distribution)). A known dataset serves as the initial training data for the discriminator. Training involves presenting it with samples from the training dataset until it achieves acceptable accuracy. The generator is trained based on whether it succeeds in fooling the discriminator. Typically, the generator is seeded with randomized input that is sampled from a predefined latent space (e.g. a multivariate normal distribution). Thereafter, candidates synthesized by the generator are evaluated by the discriminator. Independent backpropagation procedures are applied to both networks so that the generator produces better samples, while the discriminator becomes more skilled at flagging synthetic samples. When used for image generation, the generator is typically a deconvolutional neural network, and the discriminator is a convolutional neural network. Relation to other statistical machine learning methods GANs are implicit generative models, which means that they do not explicitly model the likelihood function nor provide a means for finding the latent variable corresponding to a given sample, unlike alternatives such as flow-based generative model. Compared to fully visible belief networks such as WaveNet and PixelRNN and autoregressive models in general, GANs can generate one complete sample in one pass, rather than multiple passes through the network. Compared to Boltzmann machines and nonlinear ICA, there is no restriction on the type of function used by the network. Since neural networks are universal approximators, GANs are asymptotically consistent. Variational autoencoders might be universal approximators, but it is not proven as of 2017. Mathematical properties Measure-theoretic considerations This section provides some of the mathematical theory behind these methods. In modern probability theory based on measure theory, a probability space also needs to be equipped with a σ-algebra. As a result, a more rigorous definition of the GAN game would make the following changes:Each probability space defines a GAN game. The generator's strategy set is , the set of all probability measures on the measure-space . The discriminator's strategy set is the set of Markov kernels , where is the Borel σ-algebra on .Since issues of measurability never arise in practice, these will not concern us further. Choice of the strategy set In the most generic version of the GAN game described above, the strategy set for the discriminator contains all Markov kernels , and the