We study the problem of online binary classification where strategic agents can manipulate their observable features in predefined ways, modeled by a manipulation graph, in order to receive a positive classification. We show this setting differs in fundamental ways from non-strategic online classification. For instance, whereas in the non-strategic case, a mistake bound of $\ln|H|$ is achievable via the halving algorithm when the target function belongs to a known class $H$, we show that no deterministic algorithm can achieve a mistake bound $o(\Delta)$ in the strategic setting, where $\Delta$ is the maximum degree of the manipulation graph (even when $|H|=O(\Delta)$). We obtain an algorithm achieving mistake bound $O(\Delta\ln|H|)$. We also extend this to the agnostic setting and obtain an algorithm with a $\Delta$ multiplicative regret, and we show no deterministic algorithm can achieve $o(\Delta)$ multiplicative regret. Next, we study two randomized models based on whether the random choices are made before or after agents respond, and show they exhibit fundamental differences. In the first model, at each round the learner deterministically chooses a probability distribution over classifiers inducing expected values on each vertex (probabilities of being classified as positive), which the strategic agents respond to. We show that any learner in this model has to suffer linear regret. On the other hand, in the second model, while the adversary who selects the next agent must respond to the learner's probability distribution over classifiers, the agent then responds to the actual hypothesis classifier drawn from this distribution. Surprisingly, we show this model is more advantageous to the learner, and we design randomized algorithms that achieve sublinear regret bounds against both oblivious and adaptive adversaries.

翻译：我们研究在线二分类问题，其中具有策略行为的智能体可以按照预定义方式（由操作图建模）操控其可观察特征，以获得正面分类结果。我们证明该设置与非策略性在线分类存在根本性差异。例如，在非策略场景中，当目标函数属于已知类别$H$时，通过减半算法可实现$\ln|H|$的误分类界；但在策略场景中，任何确定性算法都无法实现$o(\Delta)$的误分类界（其中$\Delta$为操作图的最大度数，即使当$|H|=O(\Delta)$时亦然）。我们提出一种误分类界为$O(\Delta\ln|H|)$的算法，并将其扩展至不可知场景，得到具有$\Delta$倍遗憾的算法，同时证明任何确定性算法都无法达到$o(\Delta)$倍遗憾。进一步地，我们基于随机选择发生在智能体响应之前或之后两种情形，研究了两种随机化模型并揭示其根本差异。在第一个模型中，每轮学习器确定性地选择分类器上的概率分布，为每个顶点生成期望值（被分类为正面的概率），策略智能体对此做出响应。我们证明该模型中任何学习器都必须承受线性遗憾。相反，在第二个模型中，虽然选择下一智能体的对手需响应学习器在分类器上的概率分布，但智能体实际响应的是从该分布中抽取的真实假设分类器。令人惊讶的是，我们证明该模型对学习器更为有利，并设计了针对无记忆性和适应性两类对手均能实现次线性遗憾界的随机化算法。