Efficient distribution-free PAC learning of k-juntas without membership queries

Determine whether k-juntas are efficiently learnable in the distribution-free PAC model without membership queries; specifically, ascertain whether there exists a distribution-free PAC learning algorithm for k-juntas whose runtime is asymptotically faster than the brute-force n^k time bound.

Background

A k-junta is a Boolean function on n variables that depends on at most k of them. The paper contrasts learning k-juntas with and without membership queries: in the exact learning and PAC+MQ settings, membership queries enable efficient identification of relevant variables and yield algorithms with runtime poly(n, 2^k). However, in the standard distribution-free PAC model without membership queries, the fastest known algorithms run in time n^k, and the authors note that improving on this bound remains unresolved.

This problem is central because many richer concept classes (e.g., DNF, decision trees) can encode juntas, so progress on junta learning often underpins broader learning advances. The authors explicitly flag the difficulty of the junta problem without membership queries as an open problem, despite partial progress in the uniform-distribution setting.