We prove a lower bound on the excess risk of sparse interpolating procedures for linear regression with Gaussian data in the overparameterized regime. We apply this result to obtain a lower bound for basis pursuit (the minimum $\ell_1$-norm interpolant) that implies that its excess risk can converge at an exponentially slower rate than OLS (the minimum $\ell_2$-norm interpolant), even when the ground truth is sparse. Our analysis exposes the benefit of an effect analogous to the "wisdom of the crowd", except here the harm arising from fitting the $\textit{noise}$ is ameliorated by spreading it among many directions -- the variance reduction arises from a $\textit{foolish}$ crowd.

翻译：事实证明,在超度参数化制度下,用高斯数据进行线性回归的内插程序稀少,其风险过多,我们用这一结果来获得较低的基础追踪限制(最低值为$_1美元-诺尔南内插),这意味着其过度风险的汇合速度可能比OSS(最低值$_2美元-诺尔南内插)快得多,即使地面真相稀少。我们的分析揭示了类似于“人群的智慧”的效果的好处,但此处除外,由于在多个方向之间扩散而使$\ textit{noise}美元所带来的伤害得到改善——差异减少产生于$textit{folish}美元人群。