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

TL;DR

This paper establishes near-optimal computational hardness results for learning homogeneous halfspaces under Gaussian distributions, extending prior results and narrowing the gap between known upper and lower bounds.

Contribution

It proves new hardness results for homogeneous halfspaces under Gaussian marginals, based on the Learning With Errors assumption, improving upon prior general halfspace results.

Findings

Hardness results are established under LWE assumption.

Results extend to homogeneous halfspaces, not just general halfspaces.

The lower bounds improve the understanding of the computational difficulty.

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.