  Maximum Mean Discrepancy (MMD) is a probability metric that has found
numerous applications in machine learning. In this work, we focus on its
application in generative models, including the minimum MMD estimator,
Generative Moment Matching Network (GMMN), and Generative Adversarial Network
(GAN). In these cases, MMD is part of an objective function in a minimization
or min-max optimization problem. Even if its empirical performance is
competitive, the consistency and convergence rate analysis of the corresponding
MMD-based estimators has yet to be carried out.
  We propose a uniform concentration inequality for a class of Maximum Mean
Discrepancy (MMD)-based estimators, that is, a maximum deviation bound of
empirical MMD values over a collection of generated distributions and
adversarially learned kernels. Here, our inequality serves as an efficient tool
in the theoretical analysis for MMD-based generative models. As elaborating
examples, we applied our main result to provide the generalization error bounds
for the MMD-based estimators in the context of the minimum MMD estimator and
MMD GAN.
