Sauer's bound for a notion of teaching complexity

Authors:
Rahim Samei;Pavel Semukhin;Boting Yang;Sandra Zilles
Affiliations:
Department of Computer Science, University of Regina, Canada;Department of Computer Science, University of Regina, Canada;Department of Computer Science, University of Regina, Canada;Department of Computer Science, University of Regina, Canada
Venue:
ALT'12 Proceedings of the 23rd international conference on Algorithmic Learning Theory
Year:
2012

Citing 7
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Abstract

This paper establishes an upper bound on the size of a concept class with given recursive teaching dimension (RTD, a teaching complexity parameter.) The upper bound coincides with Sauer's well-known bound on classes with a fixed VC-dimension. Our result thus supports the recently emerging conjecture that the combinatorics of VC-dimension and those of teaching complexity are intrinsically interlinked. We further introduce and study RTD-maximum classes (whose size meets the upper bound) and RTD-maximal classes (whose RTD increases if a concept is added to them), showing similarities but also differences to the corresponding notions for VC-dimension. Another contribution is a set of new results on maximal classes of a given VC-dimension. Methodologically, our contribution is the successful application of algebraic techniques, which we use to obtain a purely algebraic characterization of teaching sets (sample sets that uniquely identify a concept in a given concept class) and to prove our analog of Sauer's bound for RTD.