Tractable and Near-Optimal Adversarial Algorithms for Robust Estimation in Contaminated Gaussian Models
Ziyue Wang, Zhiqiang Tan; 24(235):1−112, 2023.
Abstract
This study focuses on the problem of simultaneous estimation of location and variance matrix under the Huber’s contaminated Gaussian model. The authors first investigate minimum $f$-divergence estimation at the population level, which corresponds to a generative adversarial method with a nonparametric discriminator. They establish conditions on $f$-divergences that lead to robust estimation, similar to the robustness of minimum distance estimation. Furthermore, the authors propose tractable adversarial algorithms with simple spline discriminators. These algorithms can be defined by nested optimization, where the discriminator parameters are determined by maximizing a concave objective function given the current generator. The authors demonstrate that the proposed methods achieve minimax optimal rates or near-optimal rates, depending on the $f$-divergence and the penalty used. This is the first time that such near-optimal error rates are established for adversarial algorithms with linear discriminators under the Huber’s contamination model. Simulation studies are presented to show the advantages of the proposed methods over classic robust estimators, pairwise methods, and a generative adversarial method with neural network discriminators.
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