标题： 大规模样本逻辑回归的最优子采样 显示英文标题

标题： Optimal Subsampling for Large Sample Logistic Regression

摘要： 对于大规模数据，子采样算法族因其能够减少数据量和计算负担而广受欢迎。现有研究主要集中于近似线性回归中的普通最小二乘估计，其中通常使用统计杠杆得分来定义子采样的概率。本文提出了一种快速的子采样算法，以高效地近似逻辑回归中的最大似然估计。首先，我们建立了通用子采样算法下估计量的一致性和渐近正态性，并推导出最优的子采样概率，以最小化由此产生的估计量的渐近均方误差。此外，还提出了另一种最小化准则以进一步降低计算成本。最优的子采样概率依赖于完整的数据估计值，因此我们开发了一种两步算法来近似最优的子采样过程。该算法计算效率高，并且与完整数据方法相比，计算时间有显著减少。同时，我们也证明了两步算法下估计量的一致性和渐近正态性。通过合成数据集和真实数据集验证了所提出方法的实际性能。 显示英文摘要

摘要： For massive data, the family of subsampling algorithms is popular to downsize the data volume and reduce computational burden. Existing studies focus on approximating the ordinary least squares estimate in linear regression, where statistical leverage scores are often used to define subsampling probabilities. In this paper, we propose fast subsampling algorithms to efficiently approximate the maximum likelihood estimate in logistic regression. We first establish consistency and asymptotic normality of the estimator from a general subsampling algorithm, and then derive optimal subsampling probabilities that minimize the asymptotic mean squared error of the resultant estimator. An alternative minimization criterion is also proposed to further reduce the computational cost. The optimal subsampling probabilities depend on the full data estimate, so we develop a two-step algorithm to approximate the optimal subsampling procedure. This algorithm is computationally efficient and has a significant reduction in computing time compared to the full data approach. Consistency and asymptotic normality of the estimator from a two-step algorithm are also established. Synthetic and real data sets are used to evaluate the practical performance of the proposed method.