Local Regularizers Are Not Transductive Learners
This resolves an open question in learning theory, providing a counterexample that highlights limitations of local regularization for transductive learning, with potential implications for understanding algorithmic separations.
The paper addresses whether local regularizers can learn all learnable multiclass problems, showing a negative answer in the transductive model by constructing a problem learnable in both transductive and PAC models but not by any local regularizer, using cryptographic secret sharing principles.
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.