Optimal distributed subsampling under heterogeneity

Distributed subsampling approaches have been proposed to process massive data in a distributed computing environment, where subsamples are taken from each site and then analyzed collectively to address statistical problems when the full data is not available. In this paper, we consider that each site involves a common parameter and site-specific nuisance parameters and then formulate a unified framework of optimal distributed subsampling under heterogeneity for general optimization problems with convex loss functions that could be nonsmooth. By establishing the consistency and asymptotic normality of the distributed subsample estimators for the common parameter of interest, we derive the optimal subsampling probabilities and allocation sizes under the A- and L-optimality criteria. A two-step algorithm is proposed for practical implementation and the asymptotic properties of the resultant estimator are established. For nonsmooth loss functions, an alternating direction method of multipliers method and a random perturbation procedure are proposed to obtain the subsample estimator and estimate the covariance matrices for statistical inference, respectively. The finite-sample performance of linear regression, logistic regression and quantile regression models is demonstrated through simulation studies and an application to the National Longitudinal Survey of Youth Dataset is also provided.