The Optimal Sample Complexity of Multiclass and List Learning

TL;DR

This paper establishes the exact relationship between the DS dimension and sample complexity in multiclass and list learning, resolving a longstanding open problem.

Contribution

It proves that the maximum hypergraph density is bounded by the DS dimension, confirming a conjecture and determining optimal sample complexity dependence.

Findings

Bound on hypergraph density by DS dimension

Resolution of the gap in sample complexity bounds

Confirmation of Daniely and Shalev-Shwartz's conjecture

Abstract

While the optimal sample complexity of binary classification in terms of the VC dimension is well-established, determining the optimal sample complexity of multiclass classification has remained open. The appropriate complexity parameter for multiclass classification is the DS dimension, and despite significant efforts, a gap of $\sqrt{\text{DS}}$ has persisted between the upper and lower bounds on sample complexity. Recent work by Hanneke et al. (2026) shows a novel algebraic characterization of multiclass hypothesis classes in terms of their DS dimension. Building up on this, we show that the maximum hypergraph density of any multiclass hypothesis class is upper-bounded by its DS dimension. This proves a longstanding conjecture of Daniely and Shalev-Shwartz (2014). As a consequence, we determine the optimal dependence of the sample complexity on the DS dimension for multiclass as well as list learning.