Asymptotic Behavior of Adversarial Training Estimator under
  $\ell_\infty$-Perturbation
Yiling Xie Xiaoming Huo
Adversarial training has been proposed to hedge against adversarial attacks
in machine learning and statistical models. This paper focuses on adversarial
training under $\ell_\infty$-perturbation, which has recently attracted much
research attention. The asymptotic behavior of the adversarial training
estimator is investigated in the generalized linear model. The results imply
that the limiting distribution of the adversarial training estimator under
$\ell_\infty$-perturbation could put a positive probability mass at $0$ when
the true parameter is $0$, providing a theoretical guarantee of the associated
sparsity-recovery ability. Alternatively, a two-step procedure is proposed --
adaptive adversarial training, which could further improve the performance of
adversarial training under $\ell_\infty$-perturbation. Specifically, the
proposed procedure could achieve asymptotic unbiasedness and variable-selection
consistency. Numerical experiments are conducted to show the sparsity-recovery
ability of adversarial training under $\ell_\infty$-perturbation and to compare
the empirical performance between classic adversarial training and adaptive
adversarial training.