Robust Estimation and Generative Adversarial Nets
This work addresses the problem of making robust estimation practical for statisticians and machine learning practitioners by bridging theoretical depth functions with generative adversarial networks, offering a novel computational approach.
The paper tackles the computational intractability of statistically optimal robust estimators, such as Tukey's median, by connecting them to f-GANs through f-Learning, enabling the use of GAN training tools to compute these estimators efficiently. It demonstrates theoretically and experimentally that discriminator networks in GANs can achieve statistically optimal robust location estimators for Gaussian and elliptical distributions.
Robust estimation under Huber's $ε$-contamination model has become an important topic in statistics and theoretical computer science. Statistically optimal procedures such as Tukey's median and other estimators based on depth functions are impractical because of their computational intractability. In this paper, we establish an intriguing connection between $f$-GANs and various depth functions through the lens of $f$-Learning. Similar to the derivation of $f$-GANs, we show that these depth functions that lead to statistically optimal robust estimators can all be viewed as variational lower bounds of the total variation distance in the framework of $f$-Learning. This connection opens the door of computing robust estimators using tools developed for training GANs. In particular, we show in both theory and experiments that some appropriate structures of discriminator networks with hidden layers in GANs lead to statistically optimal robust location estimators for both Gaussian distribution and general elliptical distributions where first moment may not exist.