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

翻译：我们研究了策略分类中的基本错误界和样本复杂度，其中智能体为获得正类预测结果，可在一定程度上策略性地操纵其特征向量。例如，在决定大学录取的分类器中，学生候选人可能试图选修更简单的课程以提高绩点、重考SAT或转学，以期欺骗分类器。球体操纵是文献中广泛研究的一类操纵行为，即智能体可在有界半径球内修改其特征向量。与大多数先前工作不同，本研究考虑了个性化的操纵行为，这意味着智能体可能具有不同水平的操纵能力（例如球体操纵中不同的半径范围），且这些能力对学习器而言是未知的。我们以交互模型形式化该学习问题：学习器首先部署分类器，随后智能体在其操纵集内修改特征向量以欺骗已部署的分类器。我们探究了交互过程中学习器可获取信息所对应的多种场景，包括：观察部署前后的原始特征向量、观察被操纵后的特征向量，或无法观察到原始与被操纵的特征向量。我们首先针对球体操纵给出了这些场景下的在线错误界和PAC样本复杂度。此外，我们还探讨了非球体操纵场景，并证明即便在最简单的情形（即原始与被操纵的特征向量均被揭示）下，当目标函数属于已知类别$H$时，错误界和样本复杂度均存在$\Omega(|H|)$的下界。