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

02/13/2023

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by   Ilias Diakonikolas, et al.

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We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (𝐱,y) from an unknown distribution on ℝ^n ×{± 1}, whose marginal distribution on 𝐱 is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+ϵ, where 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.