On Batch Teaching Without Collusion

Formal models of learning from teachers need to respect certain criteria to avoid collusion. The most commonly accepted notion of collusion-avoidance 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-avoidance 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 other complexity parameters of interest in computational learning theory. We further argue that Goldman and Mathias's collusion-avoidance criterion may in some set-tings be too weak in that it admits certain forms of interaction between teacher and learner that could be considered collusion in practice. Therefore, we introduce a strictly stronger notion of collusion-avoidance and demonstrate that the well-studied notion of Preference-based Teaching is optimal among all teaching schemes that are strongly collusion-avoiding on all finite subsets of a given concept class.