标题： 草图无偏差线性回归：下采样的作用 显示英文标题

标题： Sketched Ridgeless Linear Regression: The Role of Downsampling

摘要： 过参数化通常有助于提高泛化性能。 本文从双重视角探讨了过参数化，表明下采样也可能有助于泛化。 聚焦于比例区间$m\asymp n \asymp p$，其中$m$表示下采样大小，$n$是样本大小，$p$是特征维度，我们研究了下采样无偏差最小二乘估计器的两种样本外预测风险。 我们的发现挑战了传统观念，表明下采样并不总是损害泛化，而是在某些情况下实际上可以提高泛化能力。 我们确定了最小化样本外预测风险的最佳下采样大小，并证明最优下采样估计器表现出更稳定的风险曲线，消除了全样本估计器的风险峰值。 为了便于实际应用，我们提出了一种经验性方法来确定最佳下采样大小。 最后，我们将分析扩展到涵盖中心极限定理和错误设定模型。 数值研究强烈支持我们的理论。 显示英文摘要

摘要： Overparametrization often helps improve the generalization performance. This paper presents a dual view of overparametrization suggesting that downsampling may also help generalize. Focusing on the proportional regime $m\asymp n \asymp p$, where $m$ represents the sketching size, $n$ is the sample size, and $p$ is the feature dimensionality, we investigate two out-of-sample prediction risks of the sketched ridgeless least square estimator. Our findings challenge conventional beliefs by showing that downsampling does not always harm generalization but can actually improve it in certain cases. We identify the optimal sketching size that minimizes out-of-sample prediction risks and demonstrate that the optimally sketched estimator exhibits stabler risk curves, eliminating the peaks of those for the full-sample estimator. To facilitate practical implementation, we propose an empirical procedure to determine the optimal sketching size. Finally, we extend our analysis to cover central limit theorems and misspecified models. Numerical studies strongly support our theory.