We consider an active learning setting where a learner is presented with a pool S of n unlabeled examples belonging to a domain X and asks queries to find the underlying labeling that agrees with a target concept h^* \in H. In contrast to traditional active learning that queries a single example for its label, we study more general region queries that allow the learner to pick a subset of the domain T \subset X and a target label y and ask a labeler whether h^*(x) = y for every example in the set T \cap S. Such more powerful queries allow us to bypass the limitations of traditional active learning and use significantly fewer rounds of interactions to learn but can potentially lead to a significantly more complex query language. Our main contribution is quantifying the trade-off between the number of queries and the complexity of the query language used by the learner. We measure the complexity of the region queries via the VC dimension of the family of regions. We show that given any hypothesis class H with VC dimension d, one can design a region query family Q with VC dimension O(d) such that for every set of n examples S \subset X and every h^* \in H, a learner can submit O(d log n) queries from Q to a labeler and perfectly label S. We show a matching lower bound by designing a hypothesis class H with VC dimension d and a dataset S \subset X of size n such that any learning algorithm using any query class with VC dimension O(d) must make poly(n) queries to label S perfectly. Finally, we focus on well-studied hypothesis classes including unions of intervals, high-dimensional boxes, and d-dimensional halfspaces, and obtain stronger results. In particular, we design learning algorithms that (i) are computationally efficient and (ii) work even when the queries are not answered based on the learner's pool of examples S but on some unknown superset L of S

翻译：我们考虑一个主动学习场景：学习者面对一个包含 n 个未标记样本的集合 S（这些样本属于域 X），并通过查询来寻找与目标概念 h^* \in H 一致的底层标注。与传统主动学习中对单个样本进行标签查询不同，我们研究更通用的区域查询，允许学习者选择域的子集 T \subset X 和目标标签 y，并向标注者询问对于 T \cap S 中的每个样本，是否满足 h^*(x) = y。这种更强大的查询方式使我们能够绕过传统主动学习的限制，使用显著更少的交互轮次进行学习，但可能导致查询语言的复杂度显著增加。我们的主要贡献在于量化查询次数与学习者所用查询语言复杂度之间的权衡。我们通过区域族的 VC 维度来衡量区域查询的复杂度。我们证明：给定任意 VC 维度为 d 的假设类 H，可以设计一个 VC 维度为 O(d) 的区域查询族 Q，使得对于任意包含 n 个样本的集合 S \subset X 和任意 h^* \in H，学习者可以向标注者提交来自 Q 的 O(d log n) 个查询，并完美标注 S。我们通过设计一个 VC 维度为 d 的假设类 H 和一个大小为 n 的数据集 S \subset X 来证明匹配的下界：任何使用 VC 维度为 O(d) 的查询类的学习算法，都需要进行 poly(n) 次查询才能完美标注 S。最后，我们聚焦于多个被广泛研究的假设类（包括区间并集、高维盒子和 d 维半空间），并获得了更强的结果。具体而言，我们设计了满足以下条件的学习算法：(i) 计算效率高；(ii) 即使查询的答案并非基于学习者的样本池 S，而是基于 S 的某个未知超集 L，算法仍能正常工作。