Strategic Classification under Unknown Personalized Manipulation Anonymous Author(s) Affiliation Address email

We study the fundamental mistake bound and sample complexity in the strategic1 classification, where agents can strategically manipulate their feature vector up2 to an extent in order to be predicted as positive. For example, given a classifier3 determining college admission, student candidates may try to take easier classes to4 improve their GPA, retake SAT and change schools in an effort to fool the classifier.5 Ball manipulations are a widely studied class of manipulations in the literature,6 where agents can modify their feature vector within a bounded radius ball. Unlike7 most prior work, our work consider manipulations to be personalized, meaning8 that agents can have different levels of manipulation abilities (e.g., varying radii9 for ball manipulations), and unknown to the learner.10 We formalize the learning problem in an interaction model where the learner11 first deploys a classifier and the agent manipulates the feature vector within their12 manipulation set to game the deployed classifier. We investigate various scenarios13 in terms of the information available to the learner during the interaction, such14 as observing the original feature vector before or after deployment, observing the15 manipulated feature vector, or not seeing either the original or the manipulated16 feature vector. We begin by providing online mistake bounds and PAC sample17 complexity in these scenarios for ball manipulations. We also explore non-ball18 manipulations and show that, even in the simplest scenario where both the original19 and the manipulated feature vectors are revealed, the mistake bounds and sample20 complexity are lower bounded by Ω(|H|) when the target function belongs to a21 known class H.22