Necessity of finite selected ordered disagreement dimension

Determine whether finiteness of the selected ordered disagreement dimension is necessary for a regularizer to learn a multiclass hypothesis class under the hard structural risk minimization framework.

Background

The paper defines the selected ordered disagreement dimension of a regularizer by considering disagreement sets between a target hypothesis and hypotheses that can actually be selected as regularizer minimizers on some realizable sample. The authors prove that finite selected ordered disagreement dimension is sufficient for every hard-SRM selector induced by the regularizer to PAC learn the class.

They explicitly leave unresolved whether this finiteness condition is also necessary. Resolving the question would characterize hard-SRM learnability more precisely by determining whether every successful regularizer must have uniformly bounded VC complexity among its genuinely selectable alternatives.

References

Whether finiteness of the SOD dimension is necessary for $\psi$'s success, however, remains open.

— 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 Proposition 3.5 and the paragraph immediately preceding it, Section 5.2 (Ordered disagreement complexity)

Whether this additional flexibility suffices to learn every multiclass problem is an interesting direction for future work, which we leave open.

— 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 Remark 4.3, Section 4.2 (A PAC counterexample to hard local regularization)