Characterize the U-dependent sample complexity via global one-inclusion complexity

Characterize the adversarially robust PAC sample complexity for each fixed pair of concept class H and perturbation map U by determining whether the global one-inclusion complexity measure introduced by Montasser et al. is bounded above by VC(H), uniformly over all perturbation maps U.

Background

The paper establishes optimal sample-complexity rates uniformly over all perturbation maps, showing that adversarial robustness has no additional distribution-free statistical cost compared with ordinary PAC learning. However, the authors note that for a fixed pair (H, U), the statistical complexity may be substantially smaller. Prior work introduced the global one-inclusion graph and an associated combinatorial complexity measure that yields a minimax-optimal learner for each pair (H, U).

The unresolved conjecture asks whether this pair-specific complexity measure can always be controlled by the VC dimension of H, independently of U. Proving this would provide an optimal characterization of U-dependent adversarially robust PAC sample complexity, rather than only the U-uniform characterization proved in the paper.