Converse from non-insertion-stability to logarithmic penalty

Establish whether structural failure of insertion-stability necessarily forces the logarithmic monotone-insertion penalty, thereby providing a converse to the sufficient-condition result.

Background

The paper proves that if a learner is insertion-stable and has an appropriate clean-sample guarantee, then adaptive correctly labeled insertions do not worsen its risk. It also shows that intersection-closed classes admit such learners, while a particular non-intersection-closed class suffers a logarithmic penalty under monotone insertions. However, these results do not establish the converse implication for arbitrary classes or learners.

The unresolved issue is whether some structural property that prevents insertion-stability, or more generally whether sufficiently large values of the proposed insertion-stable compression dimension, must imply that every learner incurs an error rate larger than the clean O(1/n) scale. A full converse would turn the paper's one-directional sufficient condition into a characterization of the classes that suffer from monotone insertions.