Function hardness versus induced distribution hardness

Determine whether non-efficient PAC learnability of a concept class with respect to the uniform distribution and the PEX oracle implies non-efficient PAC GEN-learnability of its induced distribution class with respect to the SAMPLE oracle and total variation distance, for both classical and quantum learners.

Background

The paper considers the relationship between PAC learning Boolean concept classes and PAC generation of the induced distributions that are uniform over labeled examples. Sweke et al. established the forward implication: efficient PAC learnability of a concept class implies efficient PAC generation of its induced distribution class.

The reverse implication was proposed as a conjecture. The paper constructs, relative to an oracle encoding a one-way permutation, a concept class that is not efficiently PAC learnable but whose induced distribution class is efficiently PAC generated, thereby refuting this conjecture in the stated oracle setting. The open problem remains relevant outside that counterexample setting and under alternative assumptions or models.

References

They left the reverse direction open as a conjecture. \begin{conjecture}[Function Hardness Implies Distribution Hardness, ] If a concept class $\mathcal{C}$ is not efficiently classically (quantumly) PAC learnable with respect to the uniform distribution and the $PEX$ oracle, then the distribution class $\mathcal{D}_\mathcal{C}$ is not efficiently classically (quantumly) PAC GEN-learnable with respect to the $SAMPLE$ oracle and TV distance. \end{conjecture}

— A Quantum/Classical Example Oracle Separation for Making Things Up  (2608.11648 - Chen, 12 Aug 2026) in Conjecture 1, Section 2, subsection “Context Of Prior Work by Sweke et al.”