Improved Sample Complexity for Multiclass PAC Learning

We aim to understand the optimal PAC sample complexity in multiclass learning. While finiteness of the Daniely-Shalev-Shwartz (DS) dimension has been shown to characterize the PAC learnability of a concept class [Brukhim, Carmon, Dinur, Moran, and Yehudayoff, 2022], there exist polylog factor gaps in the leading term of the sample complexity. In this paper, we reduce the gap in terms of the dependence on the error parameter to a single log factor and also propose two possible routes towards completely resolving the optimal sample complexity, each based on a key open question we formulate: one concerning list learning with bounded list size, the other concerning a new type of shifting for multiclass concept classes. We prove that a positive answer to either of the two questions would completely resolve the optimal sample complexity up to log factors of the DS dimension.