[edit]
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 (mathbfx,y) from an unknown distribution on mathbbRntimes±1, whose marginal distribution on mathbfx is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss mathrmOPT+epsilon, where mathrmOPT 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.