DNF are teachable in the average case
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≤2Θ(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.
KeywordsDNF formulas Teaching dimension
- Alekhnovich, M., Braverman, M., Feldman, V., Klivans, A., & Pitassi, T. (2004). Learnability and automatizability. In Proceedings of the 45th IEEE symposium on foundations of computer science (pp. 621–630). Google Scholar
- Balbach, F. (2005). Teaching classes with high teaching dimension using few examples. In Proceedings of the 18th annual COLT (pp. 637–651). Google Scholar
- Blum, A. (2003). Learning a function of r relevant variables (open problem). In Proceedings of the 16th annual COLT (pp. 731–733). Google Scholar
- Cherniavsky, J., & Statman, R. (1998). Testing: An abstract approach. In Proceedings of the 2nd workshop on software testing. Google Scholar
- Kuhlmann, C. (1999). On teaching and learning intersection-closed concept classes. In Proceedings of the 4th EUROCOLT (pp. 168–182). Google Scholar
- Shinohara, A., & Miyano, S. (1990). Teachability in computational learning. In Proceedings of the first international workshop on algorithmic learning theory (pp. 247–255). Google Scholar