D-Optimal Subsampling Design for Multiple Linear Regression on Massive Data

Abstract Data reduction has become increasingly important in modern technology, as classical statistical methods are often rendered impractical by computational limitations. In the present paper, we address the situation of multiple linear regression for an extraordinarily large number of observations, but only a few covariates. Subsampling aims at the selection of a given proportion of the existing original data. Under distributional assumptions on the covariates, we derive D -optimal subsampling designs and study their theoretical properties. We make use of fundamental concepts of optimal design theory and an equivalence theorem from constrained convex optimization. The thus obtained subsampling designs provide simple rules for whether to accept or reject a data point, allowing for an easy algorithmic implementation. In addition, we propose a simplified subsampling method with lower computational complexity that deviates from the D -optimal design. We present a simulation study, comparing both subsampling schemes with the IBOSS method in the case of a fixed size of the subsample.