  An important class of distance metrics proposed for training generative
adversarial networks (GANs) is the integral probability metric (IPM), in which
the neural net distance captures the practical GAN training via two neural
networks. This paper investigates the minimax estimation problem of the neural
net distance based on samples drawn from the distributions. We develop the
first known minimax lower bound on the estimation error of the neural net
distance, and an upper bound tighter than an existing bound on the estimator
error for the empirical neural net distance. Our lower and upper bounds match
not only in the order of the sample size but also in terms of the norm of the
parameter matrices of neural networks, which justifies the empirical neural net
distance as a good approximation of the true neural net distance for training
GANs in practice.
