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：我们研究了高斯分布下半空间（halfspaces）的稳健学习（agnostic learning）问题。具体而言，给定来自 $\mathbb{R}^n \times \{ \pm 1\}$ 上未知分布的带标签样本 $(\mathbf{x},y)$，其中 $\mathbf{x}$ 的边缘分布为标准高斯分布，标签 $y$ 可以是任意的，目标是输出一个假设，使其0-1损失为 $\mathrm{OPT}+\epsilon$，其中 $\mathrm{OPT}$ 是最优拟合半空间的0-1损失。我们证明，在广泛接受的子指数时间难度假设（基于带错误学习问题，即LWE问题）下，该问题具有近似最优的计算难度。先前的难度结果要么在性质上不最优，要么仅适用于受限算法族。我们的技术可推广至相关问题（包括ReLU回归）的近似最优下界。