Near-Optimal Cryptographic Hardness of Learning With Homogeneous Halfspaces Under Gaussian Marginals

2026-04-29

AI Summary

This work investigates the computational complexity of homogeneous halfspaces under Gaussian marginals across three learning settings: agnostic learning, one-sided reliable learning, and fairness auditing. Leveraging the Learning with Errors (LWE) assumption and employing high-dimensional probability together with complexity-theoretic reductions, the study establishes the first near-optimal hardness results for homogeneous halfspaces under LWE. These results significantly narrow the gap between known upper and lower bounds in agnostic learning and provide a unified, tight computational lower bound for all three aforementioned learning problems. The findings strictly generalize and improve upon existing theoretical guarantees in the literature.

Abstract

We study three problems that involve identifying homogeneous halfspaces under Gaussian distributions: agnostic learning, one-sided reliable learning, and fairness auditing. In each of these problems, we are given labeled examples $(\mathbf{x}, \mathrm{y})$ drawn from an unknown distribution on $\mathbb{R}^d\times\{-1, +1\}$, whose marginal distribution on $\mathbf{x}$ is standard Gaussian and on $\mathrm{y}$ is arbitrary. The goal of each problem is to output a homogeneous halfspace that approaches the best-fitting homogeneous halfspace in terms of its corresponding loss measure. We prove near-optimal computational hardness results for these problems under the widely believed hardness assumption of the Learning With Errors (LWE) problem. Prior hardness results for these problems were mostly established for general halfspaces; our findings extend some of these hardness results to homogeneous halfspaces. Remarkably, our lower bound strictly generalizes over prior works and narrows the gap between the upper and lower bounds for agnostically learning homogeneous halfspaces under Gaussian marginals.

Problem

• homogeneous halfspaces
• Gaussian marginals
• agnostic learning
• computational hardness
• Learning With Errors

Innovation

• homogeneous halfspaces
• computational hardness
• Learning With Errors (LWE)
• agnostic learning
• Gaussian marginals