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.
翻译:我们研究在高斯分布法下进行非显性学习半空的工作。 具体来说, 我们给出了在 $\ mathb{ R ⁇ n\time\\ pm 1 \\\ $, 其边际分布为$\ mathbf{x} 标准高斯和标签$y 美元, 目标是输出一个假设, 损失为 0-1 $\ mathrm{ OPT ⁇ T ⁇ epsilon$, 其中$\ mathrf{ OPT} 美元是最适合的半空域的 0-1 损失 。 我们证明, 在广泛相信的 误差学习( LWE) 问题下, 其边际分布是接近最佳的计算硬性结果 。 先前的硬性结果要么是质量的次优度, 要么适用于有限的算法家族 。 我们的技术使相关问题( 包括 ReLU 回归) 产生接近最优的下界 。