  Covariate distribution shifts and adversarial perturbations present
robustness challenges to the conventional statistical learning framework: mild
shifts in the test covariate distribution can significantly affect the
performance of the statistical model learned based on the training
distribution. The model performance typically deteriorates when extrapolation
happens: namely, covariates shift to a region where the training distribution
is scarce, and naturally, the learned model has little information. For
robustness and regularization considerations, adversarial perturbation
techniques are proposed as a remedy; however, careful study needs to be carried
out about what extrapolation region adversarial covariate shift will focus on,
given a learned model. This paper precisely characterizes the extrapolation
region, examining both regression and classification in an infinite-dimensional
setting. We study the implications of adversarial covariate shifts to
subsequent learning of the equilibrium -- the Bayes optimal model -- in a
sequential game framework. We exploit the dynamics of the adversarial learning
game and reveal the curious effects of the covariate shift to equilibrium
learning and experimental design. In particular, we establish two directional
convergence results that exhibit distinctive phenomena: (1) a blessing in
regression, the adversarial covariate shifts in an exponential rate to an
optimal experimental design for rapid subsequent learning; (2) a curse in
classification, the adversarial covariate shifts in a subquadratic rate to the
hardest experimental design trapping subsequent learning.
