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

• URL: http://arxiv.org/abs/2302.06512v1
• Date: Mon, 13 Feb 2023 16:46:23 GMT
• Title: Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and ReLU Regression under Gaussian Marginals
• Authors: Ilias Diakonikolas, Daniel M. Kane, Lisheng Ren
• Abstract summary: We study the task of agnostically learning halfspaces under the Gaussian distribution. We prove a near-optimal computational hardness result for this task.
• Score: 43.0867217287089
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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