Distributed estimation in heterogeneous reduced rank regression: With application to order determination in sufficient dimension reduction

Abstract

We are concerned with massive data which are possibly heterogeneous and scattered at different locations. We introduce a communication-efficient distributed algorithm to estimate the rank-deficient loading matrix in reduced rank regressions. The distributed algorithm, which proceeds iteratively, reduces the computational complexity substantially. During each iteration, it yields a closed-form solution and refines the previous estimators gradually. After a finite number of iterations, the final solution estimates the rank consistently, and more importantly, achieves the oracle rate. We recast sufficient dimension reduction methods under the framework of reduced rank regressions, which enables us to recover the central subspace and simultaneously estimate its structural dimension. We demonstrate the efficiency of our proposed distributed algorithm through simulations and an application to the airline on-line performance dataset consisting of 118,914,458 observations.