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.

本文研究了基于概率测度的积分距离度量学习算法在神经网络距离度量的极小值估计问题。通过样本推算，我们得到了对神经网络距离度量的估计误差的极小值下限和比现有估计误差上限更紧的上界，并证明了在实践中经验神经网络距离度量是真实神经网络距离度量的有效近似。