Optimal Collusion-Free Teaching

Formal models of learning from teachers need to respect certain criteria to avoid collusion. The most commonly accepted notion of collusion-freeness was proposed by Goldman and Mathias (1996), and various teaching models obeying their criterion have been studied. For each model M and each concept class C, a parameter M-TD(C) refers to the teaching dimension of concept class C in model M---defined to be the number of examples required for teaching a concept, in the worst case over all concepts in C. This paper introduces a new model of teaching, called no-clash teaching, together with the corresponding parameter NCTD(C). No-clash teaching is provably optimal in the strong sense that, given any concept class C and any model M obeying Goldman and Mathias's collusion-freeness criterion, one obtains NCTD(C)≤ M-TD(C). We also study a corresponding notion NCTD^+ for the case of learning from positive data only, establish useful bounds on NCTD and NCTD^+, and discuss relations of these parameters to the VC-dimension and to sample compression. In addition to formulating an optimal model of collusion-free teaching, our main results are on the computational complexity of deciding whether NCTD^+(C)=k (or NCTD(C)=k) for given C and k. We show some such decision problems to be equivalent to the existence question for certain constrained matchings in bipartite graphs. Our NP-hardness results for the latter are of independent interest in the study of constrained graph matchings.