Optimal logarithmic bound for the list replicability number

Establish that every finite binary concept class with M concepts on an N-point domain has list replicability number O(log M log N), thereby proving the conjectured optimal-order bound for list replicability.

Background

For a binary concept class C with M concepts over a domain of size N, the list replicability number LR(C) is the minimum list size needed for a learner whose output, with high probability, lies in a small accurate list associated with the underlying realizable distribution. Before this note, the general finite-setting upper bound had been reduced to O(log^2 M log N), while an O(log M log N) bound had been conjectured and was known for random concept classes.

The main theorem of the paper proves only the O(log^2 M log N) bound. Thus, the stronger O(log M log N) conjecture remains unresolved in the paper; the conjectured order would be optimal up to constant factors.