  We propose a novel supervised learning method to optimize the kernel in the
maximum mean discrepancy generative adversarial networks (MMD GANs), and the
kernel support vector machines (SVMs). Specifically, we characterize a
distributionally robust optimization problem to compute a good distribution for
the random feature model of Rahimi and Recht. Due to the fact that the
distributional optimization is infinite dimensional, we consider a Monte-Carlo
sample average approximation (SAA) to obtain a more tractable finite
dimensional optimization problem. We subsequently leverage a particle
stochastic gradient descent (SGD) method to solve the derived finite
dimensional optimization problem. Based on a mean-field analysis, we then prove
that the empirical distribution of the interactive particles system at each
iteration of the SGD follows the path of the gradient descent flow on the
Wasserstein manifold. We also establish the non-asymptotic consistency of the
finite sample estimator. We evaluate our kernel learning method for the
hypothesis testing problem by evaluating the kernel MMD statistics, and show
that our learning method indeed attains better power of the test for larger
threshold values compared to an untrained kernel. Moreover, our empirical
evaluation on benchmark data-sets shows the advantage of our kernel learning
approach compared to alternative kernel learning methods.
