DNF Are Teachable in the Average Case

摘要

We study the average number of well-chosen labeled examples that are required for a helpful teacher to uniquely specify a target function within a concept class. This "average teaching dimension" has been studied in learning theory and combinatorics and is an attractive alternative to the "worst-case" teaching dimension of Goldman and Kearns which is exponential for many interesting concept classes. Recently Balbach showed that the classes of 1-decision lists and 2-term DNF each have linear average teaching dimension. As our main result, we extend Balbach's teaching result for 2-term DNF by showing that for any 1 ≤ s ≤ 2~(Θ(n)), the well-studied concept classes of at-most-s-term DNF and at-most-s-term monotone DNF each have average teaching dimension O(ns). The proofs use detailed analyses of the combinatorial structure of "most" DNF formulas and monotone DNF formulas. We also establish asymptotic separations between the worst-case and average teaching dimension for various other interesting Boolean concept classes such as juntas and sparse GF_2 polynomials.