Grinding the Space: Learning to Classify Against Strategic Agents
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We study the problem of online learning in strategic classification settings from the perspective of the learner, who is repeatedly facing myopically rational strategic agents. We model this interplay as a repeated Stackelberg game, where at each timestep the learner deploys a high-dimensional linear classifier first and an agent, after observing the classifier, along with his real feature vector, and according to his underlying utility function, best-responds with a (potentially altered) feature vector. We measure the performance of the learner in terms of Stackelberg regret for her 0-1 loss function. Surprisingly, we prove that in strategic settings like the one considered in this paper there exist worst-case scenarios, where any sequence of actions providing sublinear external regret might result in linear Stackelberg regret and vice versa. We then provide the Grinder Algorithm, an adaptive discretization algorithm, potentially of independent interest in the online learning community, and prove its data-dependent upper bound on the Stackelberg regret given oracle access, while being computationally efficient. We also provide a nearly matching lower bound for the problem of strategic classification. We complement our theoretical analysis with simulation results, which suggest that our algorithm outperforms the benchmarks, even given access to approximation oracles. Our results advance the known state-of-the-art results in the growing literature of online learning from revealed preferences, which has so far focused on smoother utility and loss functions from the perspective of the agents and the learner respectively.
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