D-optimal Subsampling Design for Massive Data Linear Regression

Abstract: Data reduction is a fundamental challenge of modern technology, where classical statistical methods are not applicable because of computational limitations. We consider 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.