Online Learning with Set-Valued Feedback
Vinod Raman, Unique Subedi, Ambuj Tewari
TL;DR
The paper introduces online learning with set-valued feedback, where the learner predicts a single label but receives a set of correct labels $S_t$ and pays a loss when $\hat{y}_t \notin S_t$. It defines two key dimensions, the Set Littlestone dimension $\text{SL}(\mathcal{H})$ for deterministic realizable learnability and the Measure Shattering dimension $\text{MS}_{\gamma}(\mathcal{H})$ for randomized realizable and agnostic learnability, with a central role for the Helly number $H(\mathcal{S}(\mathcal{Y}))$ in relating these dimensions. The results show a separation between deterministic and randomized learnability in the realizable setting (unless $H(\mathcal{S}(\mathcal{Y}))<\infty$), provide algorithmic realizations (deterministic SOA and randomized chaining-based methods), and extend to agnostic learnability with minimax regret bounds. Applications to online multilabel ranking, online multilabel classification, and interval-valued prediction illustrate the practical impact of the theory by yielding sharp, dimension-dependent minimax guarantees.
Abstract
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 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.