We study a variant of online multiclass classification where the learner predicts a single label but receives a \textit{set of labels} as feedback. In this model, the learner is penalized for not outputting a label contained in the revealed set. We show that unlike online multiclass learning with single-label feedback, deterministic and randomized online learnability are \textit{not equivalent} even in the realizable setting with set-valued feedback. Accordingly, we give two new combinatorial dimensions, named the Set Littlestone and Measure Shattering dimension, that tightly characterize deterministic and randomized online learnability respectively in the realizable setting. In addition, we show that the Measure Shattering dimension tightly characterizes online learnability in the agnostic setting. Finally, we show that practical learning settings like online multilabel ranking, online multilabel classification, and online interval learning are specific instances of our general framework.

翻译：我们研究了一种在线多类分类的变体，其中学习器预测单个标签，但接收一个标签集合作为反馈。在该模型中，若学习器输出的标签不在所揭示的集合中，则受到惩罚。我们证明，与单标签反馈的在线多类学习不同，即使在可实现情境下，确定性在线学习与随机化在线学习在集合反馈下也是不等价的。据此，我们提出了两个新的组合维度，分别命名为集合Littlestone维和测度分裂维，它们在可实现情境下分别严格刻画了确定性在线学习和随机化在线学习。此外，我们证明测度分裂维在不可知情境下严格刻画了在线学习能力。最后，我们展示在线多标签排序、在线多标签分类和在线区间学习等实际学习场景，均是我们通用框架的具体实例。