Online Learning with Set-Valued Feedback

We study a variant of online multiclass classification where the learner predicts a single label but receives a $\mathit{\text{set}}\mathit{\text{of}}\mathit{\text{labels}}$ 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 $\mathit{\text{not}}\mathit{\text{equivalent}}$ 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 characterizes online learnability in the agnostic setting and tightly quantifies the minimax regret. Finally, we use our results to establish bounds on the minimax regret for three practical learning settings: online multilabel ranking, online multilabel classification, and real-valued prediction with interval-valued response.