PAC Learning with Query Feedback

We study realizable multiclass PAC learning with query feedback, where each training example provides a single binary response indicating whether its true label belongs to a chosen subset from a fixed query family. Our main contribution is the query dimension, a combinatorial parameter that captures both the structure of the hypothesis class and the ability of the available queries to distinguish labels. This parameter generalizes the bandit DS dimension, which it recovers under singleton queries. We establish a sample-complexity lower bound of $Ω(D/\varepsilon)$ and an upper bound of $\widetilde{O}(D^2/\varepsilon)$, suppressing factors polylogarithmic in the number of labels and the failure probability. When the available queries can split every candidate set into two parts of comparable size, we improve the upper bound to $\widetilde{O}(D/\varepsilon)$, yielding a characterization up to logarithmic factors.