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Consider the problem of simultaneous estimation of location and variancematrix under Huber's contaminated Gaussian model. First, we study minimum$f$-divergence estimation at the population level, corresponding to agenerative adversarial method with a nonparametric discriminator and establishconditions on $f$-divergences which lead to robust estimation, similarly torobustness of minimum distance estimation. More importantly, we developtractable adversarial algorithms with simple spline discriminators, which canbe implemented via nested optimization such that the discriminator parameterscan be fully updated by maximizing a concave objective function given thecurrent generator. The proposed methods are shown to achieve minimax optimalrates or near-optimal rates depending on the $f$-divergence and the penaltyused. This is the first time such near-optimal error rates are established foradversarial algorithms with linear discriminators under Huber's contaminationmodel. We present simulation studies to demonstrate advantages of the proposedmethods over classic robust estimators, pairwise methods, and a generativeadversarial method with neural network discriminators.

Tractable and Near-Optimal Adversarial Algorithms for Robust Estimation in Contaminated Gaussian Models