Extend query-to-sample reductions to adaptive query algorithms

Establish whether the query-to-sample reduction framework for PAC-verification of agnostic learning can be extended to settings where the verifier’s query construction is adaptive, including adaptivity that may depend on interaction with the prover; characterize the necessary conditions (e.g., embeddability and marginal distributions) under which such adaptive reductions are possible and sound.

Background

The paper develops a query-to-sample reduction that lets a verifier replace membership queries with random examples under certain non-adaptive query-generation patterns and marginal distribution conditions. This reduction underpins the sample-efficient interactive protocols for learning heavy Fourier characters and for circuit classes via NW reconstruction.

However, the current framework relies on non-adaptivity to embed a random example into the verifier’s query set. Extending this method to adaptive query-generation—where each query can depend on prior answers or on interaction—would broaden its applicability and potentially yield stronger PAC-verification protocols.