We study the problem of PAC learning $\gamma$-margin halfspaces with Random Classification Noise. We establish an information-computation tradeoff suggesting an inherent gap between the sample complexity of the problem and the sample complexity of computationally efficient algorithms. Concretely, the sample complexity of the problem is $\widetilde{\Theta}(1/(\gamma^2 \epsilon))$. We start by giving a simple efficient algorithm with sample complexity $\widetilde{O}(1/(\gamma^2 \epsilon^2))$. Our main result is a lower bound for Statistical Query (SQ) algorithms and low-degree polynomial tests suggesting that the quadratic dependence on $1/\epsilon$ in the sample complexity is inherent for computationally efficient algorithms. Specifically, our results imply a lower bound of $\widetilde{\Omega}(1/(\gamma^{1/2} \epsilon^2))$ on the sample complexity of any efficient SQ learner or low-degree test.

翻译：我们研究了带随机分类噪声的PAC学习$\gamma$-边际半空间问题。我们建立了一个信息-计算权衡，表明该问题的样本复杂度与计算高效算法的样本复杂度之间存在固有差距。具体而言，问题的样本复杂度为$\widetilde{\Theta}(1/(\gamma^2 \epsilon))$。首先，我们给出一个简单高效的算法，其样本复杂度为$\widetilde{O}(1/(\gamma^2 \epsilon^2))$。我们的主要结果是针对统计查询(SQ)算法和低次多项式检验的下界，表明样本复杂度对$1/\epsilon$的二次依赖性对于计算高效算法是固有的。特别地，我们的结果意味着任何高效SQ学习器或低次检验的样本复杂度下界为$\widetilde{\Omega}(1/(\gamma^{1/2} \epsilon^2))$。