Complexity theoretic hardness results for query learning

Abstract

We investigate the complexity of learning for the well-studied model in which the learning algorithm may ask membership and equivalence queries. While complexity theoretic techniques have previously been used to prove hardness results in various learning models, these techniques typically are not strong enough to use when a learning algorithm may make membership queries. We develop a general technique for proving hardness results for learning with membership and equivalence queries (and for more general query models). We apply the technique to show that, assuming \( {\rm NP} \neq \hbox {\rm co-NP} \), no polynomial-time membership and (proper) equivalence query algorithms exist for exactly learning read-thrice DNF formulas, unions of \( k \ge 3 \) halfspaces over the Boolean domain, or some other related classes. Our hardness results are representation dependent, and do not preclude the existence of representation independent algorithms.¶The general technique introduces the representation problem for a class F of representations (e.g., formulas), which is naturally associated with the learning problem for F. This problem is related to the structural question of how to characterize functions representable by formulas in F, and is a generalization of standard complexity problems such as Satisfiability. While in general the representation problem is in \( \sum^{\rm P}_2 \), we present a theorem demonstrating that for "reasonable" classes F, the existence of a polynomial-time membership and equivalence query algorithm for exactly learning F implies that the representation problem for F is in fact in co-NP. The theorem is applied to prove hardness results such as the ones mentioned above, by showing that the representation problem for specific classes of formulas is NP-hard.