🤖 AI Summary
For massive-scale multivariate linear regression where covariate dimensionality is low but the number of observations is extremely large, this paper proposes a subsampling design method based on the D-optimality criterion. Under mild assumptions on the covariate distribution, we integrate optimal experimental design theory with an equivalence theorem for constrained convex optimization to derive an analytically tractable acceptance–rejection rule; we further develop a computationally lightweight approximation algorithm. This work is the first to systematically embed optimal design theory into the subsampling framework for large-scale regression, substantially improving both statistical efficiency and computational speed. Simulation studies demonstrate superior performance over IBOSS. The resulting subsample-based estimator achieves asymptotic information-theoretic optimality, balancing estimation accuracy, robustness to model misspecification, and scalability to ultra-high-volume data.
📝 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.