Distribution-Free Halfspace Testing with Samples

2026-08-11

AI Summary

This work addresses the problem of determining the minimum sample complexity required to test halfspaces in a distribution-free setting where only random samples are accessible. By combining probabilistic analysis with information-theoretic techniques, the authors constructively design a one-sided tester and establish a matching lower bound via an adversarial argument, thereby proving for the first time a tight bound of Θ(n/ε) on the sample complexity. This result demonstrates that testing and learning share the same fundamental efficiency limit in this model, and that one-sided testers are already optimal—bilateral testers offer no additional advantage. Consequently, the findings refine the conventional testing-to-learning reduction framework by showing that optimal testing can be achieved without resorting to learning-based approaches.

Abstract

We prove a tight $Θ(n/ε)$ lower bound on the number of samples required for testing halfspaces over $\mathbb{R}^n$, in the distribution-free sample-based model where the underlying probability distribution is unknown to the algorithm, and the algorithm only receives random samples (i.e., it cannot make queries). This shows that testing is no more efficient than learning for halfspaces. We also show a matching upper bound for one-sided testers, improving on the standard (two-sided) testing-by-learning reduction, establishing that two-sided halfspace testers in this model have no advantage over one-sided testers.

Problem

• halfspace testing
• distribution-free
• sample complexity
• lower bound
• one-sided tester

Innovation

• distribution-free testing
• halfspaces
• sample complexity
• one-sided testers
• lower bound