Extension of weighted-SRM integrability to infinite systems

Establish whether the finite weighted revealed-preference cycle criterion extends to infinite hypothesis systems and sample families under suitable compactness assumptions.

Background

For finite hypothesis classes and finite collections of samples, the paper proves an exact characterization of weighted-SRM representability: a learner's choices admit representation by one regularizer precisely when every directed cycle in the associated weighted revealed-preference multigraph has strictly positive total weight.

The authors suggest extending this characterization beyond finite systems by using compactness assumptions, but do not establish such a theorem. The unresolved issue is therefore whether positivity of all weighted preference cycles remains sufficient for constructing a regularizer in infinite settings, and what compactness hypotheses are required.

References

We suspect that it should be possible to extend this cycle criterion to infinite systems, by invoking certain compactness assumptions, but that remains an interesting direction for future work.

— Algorithmic Principles For Multiclass Learning Are Hard To Come By: Limits of Regularization and Proper Learning  (2608.26516 - Asilis et al., 27 Aug 2026) in Paragraph immediately preceding Definition 5.6, Section 5.3 (Integrability of revealed preferences)