We address the question of repeatedly learning linear classifiers against agents who are strategically trying to game the deployed classifiers, and we use the Stackelberg regret to measure the performance of our algorithms. First, we show that Stackelberg and external regret for the problem of strategic classification are strongly incompatible: i.e., there exist worst-case scenarios, where any sequence of actions providing sublinear external regret might result in linear Stackelberg regret and vice versa. Second, we present a strategy-aware algorithm for minimizing the Stackelberg regret for which we prove nearly matching upper and lower regret bounds. Finally, we provide simulations to complement our theoretical analysis. Our results advance the growing literature of learning from revealed preferences, which has so far focused on "smoother" assumptions from the perspective of the learner and the agents respectively.

Given such a setup, a natural question to ask is how does one measure the performance of the learner? Classical online learning studies one such notion of performance known as regret.

The experiments to produce the figures of the paper were performed on a Linux-based (CentOS) desktop with Intel Xeon CPU E5-2643 v4 @ 3.40GHz (6 cores) and 128 GB of RAM. To produce Figs. 4 and 5 (also supporting Figs. Requirements: python, matplotlib, numpy, scipy The hyperparameters obtained with optuna [2] are provided in the subdirectory results. To recreate a figure run the corresponding script. Bio-OFC is an instance of such a controller.