Revisiting the Learnability of Apple Tasting

In online binary classification under \textit{apple tasting} feedback, the learner only observes the true label if it predicts "1". First studied by \cite{helmbold2000apple}, we revisit this classical partial-feedback setting and study online learnability from a combinatorial perspective. We show that the Littlestone dimension continues to prove a tight quantitative characterization of apple tasting in the agnostic setting, closing an open question posed by \cite{helmbold2000apple}. In addition, we give a new combinatorial parameter, called the Effective width, that tightly quantifies the minimax expected mistakes in the realizable setting. As a corollary, we use the Effective width to establish a \textit{trichotomy} of the minimax expected number of mistakes in the realizable setting. In particular, we show that in the realizable setting, the expected number of mistakes for any learner under apple tasting feedback can only be $\Theta(1), \Theta(\sqrt{T})$, or $\Theta(T)$.

Multiclass Online Learnability under Bandit Feedback

We study online multiclass classification under bandit feedback. We extend the results of (daniely2013price) by showing that the finiteness of the Bandit Littlestone dimension is necessary and sufficient for bandit online multiclass learnability even when the label space is unbounded. Our result complements the recent work by (hanneke2023multiclass) who show that the Littlestone dimension characterizes online multiclass learnability in the full-information setting when the label space is unbounded.