On Teaching and Learning Intersection-Closed Concept Classes

Abstract

We consider the self-directed learning model [7] which is a variant of Littlestone's mistake-bound model [9,10]. We will refute the conjecture of [8,2] that for intersection-closed concept classes, the selfdirected learning complexity is related to the VC-dimension. We show that, even under the assumption of intersection-closedness, both parameters are completely incomparable. We furthermore investigate the structure of intersection-closed concept classes which are difficult to learn in the self-directed learning model. We show that such classes must contain maximum classes. We consider the teacher-directed learning model [5] in the worst, best and average case performance. While the teaching complexity in the worst case is incomparable to the VC-dimension, large concept classes (e.g. balls) are bounded by VC-dimension with respect to the average case. We show that the teaching complexity in the best case is bounded by the self-directed learning complexity. It is also bounded by the VC-dimension, if the concept class is intersection-closed. This does not hold for arbitrary concept classes. We find examples which substantiate this gap.