Recursive Teaching Dimension, Learning Complexity, and Maximum Classes

Abstract

This paper is concerned with the combinatorial structure of concept classes that can be learned from a small number of examples. We show that the recently introduced notion of recursive teaching dimension (RTD, reflecting the complexity of teaching a concept class) is a relevant parameter in this context. Comparing the RTD to self-directed learning, we establish new lower bounds on the query complexity for a variety of query learning models and thus connect teaching to query learning.

For many general cases, the RTD is upper-bounded by the VC-dimension, e.g., classes of VC-dimension 1, (nested differences of) intersection-closed classes, “standard” boolean function classes, and finite maximum classes. The RTD thus is the first model to connect teaching to the VC-dimension.

The combinatorial structure defined by the RTD has a remarkable resemblance to the structure exploited by sample compression schemes and hence connects teaching to sample compression. Sequences of teaching sets defining the RTD coincide with unlabeled compression schemes both (i) resulting from Rubinstein and Rubinstein’s corner-peeling and (ii) resulting from Kuzmin and Warmuth’s Tail Matching algorithm.

This work was supported by the Deutsche Forschungsgemeinschaft Grant SI 498/8-1 and by the Natural Sciences and Engineering Research Council of Canada (NSERC).