We study the problem of learning general (i.e., not necessarily homogeneous) halfspaces with Random Classification Noise under the Gaussian distribution. We establish nearly-matching algorithmic and Statistical Query (SQ) lower bound results revealing a surprising information-computation gap for this basic problem. Specifically, the sample complexity of this learning problem is $\widetilde{\Theta}(d/\epsilon)$, where $d$ is the dimension and $\epsilon$ is the excess error. Our positive result is a computationally efficient learning algorithm with sample complexity $\tilde{O}(d/\epsilon + d/(\max\{p, \epsilon\})^2)$, where $p$ quantifies the bias of the target halfspace. On the lower bound side, we show that any efficient SQ algorithm (or low-degree test) for the problem requires sample complexity at least $\Omega(d^{1/2}/(\max\{p, \epsilon\})^2)$. Our lower bound suggests that this quadratic dependence on $1/\epsilon$ is inherent for efficient algorithms.

翻译：我们研究了在随机分类噪声下学习一般（即非必须齐次）半空间问题，针对高斯分布。我们建立了近乎匹配的算法和统计查询下界结果，揭示了这一基本问题中令人惊讶的信息-计算差距。具体而言，该学习问题的样本复杂度为$\widetilde{\Theta}(d/\epsilon)$，其中$d$是维度，$\epsilon$是超额误差。我们的正向结果是提出一种计算高效的学习算法，其样本复杂度为$\tilde{O}(d/\epsilon + d/(\max\{p, \epsilon\})^2)$，其中$p$量化了目标半空间的偏移量。在下界方面，我们证明任何针对该问题的高效SQ算法（或低次检验）所需的样本复杂度至少为$\Omega(d^{1/2}/(\max\{p, \epsilon\})^2)$。我们的下界表明，对高效算法而言，这种关于$1/\epsilon$的二次依赖是固有的。