Blessings and Curses of Covariate Shifts: Adversarial Learning Dynamics, Directional Convergence, and Equilibria

TL;DR

This paper analyzes how covariate shifts and adversarial perturbations affect model robustness, revealing both beneficial and detrimental effects on learning dynamics and equilibrium in regression and classification tasks.

Contribution

It provides a precise characterization of extrapolation regions under covariate shifts and studies the resulting learning dynamics within a game-theoretic framework.

Findings

In regression, adversarial shifts lead to rapid convergence to optimal experimental design.

In classification, adversarial shifts cause slower, trapping convergence to difficult designs.

The study uncovers directional convergence phenomena with contrasting effects in regression and classification.

Abstract

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.