A New Abstract Combinatorial Dimension for Exact Learning via Queries

Abstract

We introduce an abstract model of exact learning via queries that can be instantiated to all the query learning models currently in use, while being closer to them than previous unifying attempts. We present a characterization of those Boolean function classes learnable in this abstract model, in terms of a new combinatorial notion that we introduce, the abstract identification dimension. Then we prove that the particularization of our notion to specific known protocols such as equivalence, membership, and membership and equivalence queries results in exactly the same combinatorial notions currently known to characterize learning in these models, such as strong consistency dimension, extended teaching dimension, and certificate size. Our theory thus fully unifies all these characterizations. For models enjoying a specific property that we identify, the notion can be simplified while keeping the same characterizations. From our results we can derive combinatorial characterizations of all those other models for query learning proposed in the literature. We can also obtain the first polynomial-query learning algorithms for specific interesting problems such as learning DNF with proper subset and superset queries.