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# Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and ReLU Regression under Gaussian Marginals

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:7922-7938, 2023.

#### Abstract

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\\mathbf{x},y)$ from an unknown distribution on $\\mathbb{R}^n \\times \\{\pm 1 \\}$, whose marginal distribution on $\\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\\mathrm{OPT}+\\epsilon$, where $\\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.