Given data ${\rm X}\in\mathbb{R}^{n\times d}$ and labels $\mathbf{y}\in\mathbb{R}^{n}$ the goal is find $\mathbf{w}\in\mathbb{R}^d$ to minimize $\Vert{\rm X}\mathbf{w}-\mathbf{y}\Vert^2$. We give a polynomial algorithm that, \emph{oblivious to $\mathbf{y}$}, throws out $n/(d+\sqrt{n})$ data points and is a $(1+d/n)$-approximation to optimal in expectation. The motivation is tight approximation with reduced label complexity (number of labels revealed). We reduce label complexity by $\Omega(\sqrt{n})$. Open question: Can label complexity be reduced by $\Omega(n)$ with tight $(1+d/n)$-approximation?

翻译：给定数据${\rm X}\in\mathbb{R}^{n\times d}$和标签$\mathbf{y}\in\mathbb{R}^{n}$，目标是找到$\mathbf{w}\in\mathbb{R}^d$以最小化$\Vert{\rm X}\mathbf{w}-\mathbf{y}\Vert^2$。我们提出一种多项式时间算法，该算法独立于$\mathbf{y}$，舍弃$n/(d+\sqrt{n})$个数据点，并在期望意义下达到最优解的$(1+d/n)$-近似。其动机是在降低标签复杂度（即揭示的标签数量）的同时实现紧致近似。我们将标签复杂度降低了$\Omega(\sqrt{n})$个量级。开放性问题：能否在保持$(1+d/n)$-紧致近似的前提下，将标签复杂度降低$\Omega(n)$个量级？