DNF Are Teachable in the Average Case

Abstract

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 [7] which is exponential for many interesting concept classes. Recently Balbach [3] 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 $^{\Theta({\it n})}$ , 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.