Learning Monotone DNF from a Teacher That Almost Does Not Answer Membership Queries

Abstract

We present results concerning the learning of Monotone DNF (MDNF) from Incomplete Membership Queries and Equivalence Queries. Our main result is a new algorithm that allows efficient learning of MDNF using Equivalence Queries and Incomplete Membership Queries with probability of $p = 1 - 1/poly\left( {n,t} \right)$ of failing. Our algorithm is expected to make

$$ 0\left( {\left( {\frac{{tn}} {{1 - p}}} \right)^2 } \right) $$

queries, when learning a MDNF formula with t terms over n variables. Note that this is polynomial for any failure probability p = 1-1/poly(n, t). The algorithm’s running time is also polynomial in t, n, and 1/(1-{tip}). In a sense this is the best possible, as learning with p = 1-1/ω(poly(n, t)) would imply learning MDNF, and thus also DNF, from equivalence queries alone.