A Note on Teaching for VC Classes

where we use c |X to denote the projection of c on X. The teaching dimension of C is the smallest number t such that every c ∈ C has a teaching set of size no more than t [GK95]. However, teaching dimension does not always capture the cooperation in teaching and learning, and the notion of recursive teaching dimension has been introduced and studied extensively in the literature [Kuh99, DSZ10, ZLHZ11, WY12, DFSZ14, SSYZ14, MSWY15]. The recursive teaching dimension of a class C ⊆ {0, 1}, denoted by RTD(C), is the smallest number t where one can order all the concepts of C as a sequence c1, . . . , c|C| such that every concept ci, i < |C|, has a teaching set of size no more than t in {ci, . . . , c|C|}. Hence, RTD(C) measures the worst-case number of labelled examples needed to learn any target concept in C, if the teacher and the learner agree a priori on a specific order of the concepts of the class C. In this note, we study the recursive teaching dimension of concept classes of low VC-dimension. Recall that the VC-dimension [VC71] of C ⊆ {0, 1}, denoted by VCD(C), is the maximum size of a shattered subset of [n], where Y ⊆ [n] is shattered if for every binary string b of length |Y |, there is a concept c ∈ C such that c |Y = b. Our main result is the following upper bound for RTD(C).