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.

Support vector machine (SVM) is a well-known statistical technique for classification problems in machine learning and other fields.An important question for SVM is the selection of covariates (or features) for the model.Many studies have considered model selection methods. As is well-known, selecting one winning model over others can entail considerable instability in predictive performance due to model selection uncertainties. This paper advocates model averaging as an alternative approach, where estimates obtained from different models are combined in a weighted average. We propose a model weighting scheme and provide the theoretical underpinning for the proposed method. In particular, we prove that our proposed method yields a model average estimator that achieves the smallest hinge risk among all feasible combinations asymptotically. To remedy the computational burden due to a large number of feasible models, we propose a screening step to eliminate the uninformative features before combining the models. Results from real data applications and a simulation study show that the proposed method generally yields more accurate estimates than existing methods.