Local Regularizers Are Not Transductive Learners

TL;DR

This paper demonstrates that local regularizers, a form of structural risk minimization, are insufficient for transductive learning in certain multiclass problems, highlighting limitations of this approach.

Contribution

It provides the first negative result showing local regularizers cannot learn some problems in the transductive setting, using cryptographic principles.

Findings

Identifies a multiclass problem learnable in transductive and PAC models but not by local regularizers.

Uses cryptographic secret sharing principles in the proof.

Highlights a potential separation between PAC and transductive learning with local regularizers.

Abstract

We partly resolve an open question raised by Asilis et al. (COLT 2024): whether the algorithmic template of local regularization -- an intriguing generalization of explicit regularization, a.k.a. structural risk minimization -- suffices to learn all learnable multiclass problems. Specifically, we provide a negative answer to this question in the transductive model of learning. We exhibit a multiclass classification problem which is learnable in both the transductive and PAC models, yet cannot be learned transductively by any local regularizer. The corresponding hypothesis class, and our proof, are based on principles from cryptographic secret sharing. We outline challenges in extending our negative result to the PAC model, leaving open the tantalizing possibility of a PAC/transductive separation with respect to local regularization.