**M**y friend Gérard Biau and his coauthors have published a paper in the Annals of Statistics last year on the theoretical [statistical] analysis of GANs, which I had missed and recently read with a definitive interest in the issues. (With no image example!)

If the discriminator is unrestricted the unique optimal solution is the Bayes posterior probability

when the model density is everywhere positive. And the optimal parameter θ corresponds to the closest model in terms of Kullback-Leibler divergence. The pseudo-true value of the parameter. This is however the ideal situation, while in practice D is restricted to a parametric family. In this case, if the family is wide enough to approximate the ideal discriminator in the sup norm, with error of order ε, and if the parameter space Θ is compact, the optimal parameter found under the restricted family approximates the pseudo-true value in the sense of the GAN loss, at the order ε². With a stronger assumption on the family ability to approximate any discriminator, the same property holds for the empirical version (and in expectation). (As an aside, the figure illustrating this property confusedly uses an histogramesque rectangle to indicate the expectation of the discriminator loss!) And both parameter (θ and α) estimators converge to the optimal ones with the sample size. An interesting foray from statisticians in a method whose statistical properties are rarely if ever investigated. Missing a comparison with alternative approaches, like MLE, though.