  This paper studies generative adversarial networks (GANs) from a statistical
perspective. A GAN is a popular machine learning method in which the parameters
of two neural networks, a generator and a discriminator, are estimated to solve
a particular minimax problem. This minimax problem typically has a multitude of
solutions and the focus of this paper are the statistical properties of these
solutions. We address two key issues for the generator and discriminator
network parameters, consistent estimation and confidence sets. We first show
that the set of solutions to the sample GAN problem is a (Hausdorff) consistent
estimator of the set of solutions to the corresponding population GAN problem.
We then devise a computationally intensive procedure to form confidence sets
and show that these sets contain the population GAN solutions with the desired
coverage probability. The assumptions employed in our results are weak and hold
in many practical GAN applications. To the best of our knowledge, this paper
provides the first results on statistical inference for GANs in the empirically
relevant case of multiple solutions.
