Optimal Subsampling Approaches for Large Sample Linear Regression
This work addresses computational bottlenecks in large-scale statistical inference for researchers and practitioners, presenting incremental improvements over existing subsampling techniques.
The paper tackles the challenge of analyzing large sample data by developing optimal subsampling methods for linear regression, introducing weighted and unweighted algorithms with the weighted one achieving consistency and the unweighted one showing better coefficient estimation performance in simulations.
A significant hurdle for analyzing large sample data is the lack of effective statistical computing and inference methods. An emerging powerful approach for analyzing large sample data is subsampling, by which one takes a random subsample from the original full sample and uses it as a surrogate for subsequent computation and estimation. In this paper, we study subsampling methods under two scenarios: approximating the full sample ordinary least-square (OLS) estimator and estimating the coefficients in linear regression. We present two algorithms, weighted estimation algorithm and unweighted estimation algorithm, and analyze asymptotic behaviors of their resulting subsample estimators under general conditions. For the weighted estimation algorithm, we propose a criterion for selecting the optimal sampling probability by making use of the asymptotic results. On the basis of the criterion, we provide two novel subsampling methods, the optimal subsampling and the predictor- length subsampling methods. The predictor-length subsampling method is based on the L2 norm of predictors rather than leverage scores. Its computational cost is scalable. For unweighted estimation algorithm, we show that its resulting subsample estimator is not consistent to the full sample OLS estimator. However, it has better performance than the weighted estimation algorithm for estimating the coefficients. Simulation studies and a real data example are used to demonstrate the effectiveness of our proposed subsampling methods.