  Consider the problem of simultaneous estimation of location and variance
matrix under Huber's contaminated Gaussian model. First, we study minimum
$f$-divergence estimation at the population level, corresponding to a
generative adversarial method with a nonparametric discriminator and establish
conditions on $f$-divergences which lead to robust estimation, similarly to
robustness of minimum distance estimation. More importantly, we develop
tractable adversarial algorithms with simple spline discriminators, which can
be implemented via nested optimization such that the discriminator parameters
can be fully updated by maximizing a concave objective function given the
current generator. The proposed methods are shown to achieve minimax optimal
rates or near-optimal rates depending on the $f$-divergence and the penalty
used. This is the first time such near-optimal error rates are established for
adversarial algorithms with linear discriminators under Huber's contamination
model. We present simulation studies to demonstrate advantages of the proposed
methods over classic robust estimators, pairwise methods, and a generative
adversarial method with neural network discriminators.
