Subsampling is effective in tackling computational challenges for massive data with rare events. Overly aggressive subsampling may adversely affect estimation efficiency, and optimal subsampling is essential to mitigate the information loss. However, existing optimal subsampling probabilities depends on data scales, and some scaling transformations may result in inefficient subsamples. This problem is more significant when there are inactive features, because their influence on the subsampling probabilities can be arbitrarily magnified by inappropriate scaling transformations. We tackle this challenge and introduce a scale-invariant optimal subsampling function in the context of sparse models, where inactive features are commonly assumed. Instead of focusing on estimating model parameters, we define an optimal subsampling function to minimize the prediction error, using adaptive lasso as an example to outline the estimation procedure and study its theoretical guarantee. We first introduce the adaptive lasso estimator for rare-events data and establish its oracle properties, thereby validating the use of subsampling. Then we derive a scale-invariant optimal subsampling function that minimizes the prediction error of the inverse probability weighted (IPW) adaptive lasso. Finally, we present an estimator based on the maximum sampled conditional likelihood (MSCL) to further improve the estimation efficiency. We conduct numerical experiments using both simulated and real-world data sets to demonstrate the performance of the proposed methods.

Faced with massive data, subsampling is a commonly used technique to improve computational efficiency, and using nonuniform subsampling probabilities is an effective approach to improve estimation efficiency. For computational efficiency, subsampling is often implemented with replacement or through Poisson subsampling. However, no rigorous investigation has been performed to study the difference between the two subsampling procedures such as their estimation efficiency and computational convenience. This paper performs a comparative study on these two different sampling procedures. In the context of maximizing a general target function, we first derive asymptotic distributions for estimators obtained from the two sampling procedures. The results show that the Poisson subsampling may have a higher estimation efficiency. Based on the asymptotic distributions for both subsampling with replacement and Poisson subsampling, we derive optimal subsampling probabilities that minimize the variance functions of the subsampling estimators. These subsampling probabilities further reveal the similarities and differences between subsampling with replacement and Poisson subsampling. The theoretical characterizations and comparisons on the two subsampling procedures provide guidance to select a more appropriate subsampling approach in practice. Furthermore, practically implementable algorithms are proposed based on the optimal structural results, which are evaluated through both theoretical and empirical analyses.