Distributed subsampling for multiplicative regression

Multiplicative regression is a useful alternative tool in modeling positive response data. This paper proposes two distributed estimators for multiplicative error model on distributed system with non-randomly distributed massive data. We first present a Poisson subsampling procedure to obtain a subsampling estimator based on the least product relative error (LPRE) loss, which is effective on a distributed system. Theoretically, we justify the subsampling estimator by establishing its convergence rate, asymptotic normality and deriving the optimal subsampling probabilities in terms of the L-optimality criterion. Then, we provide a distributed LPRE estimator based on the Poisson subsampling (DLPRE-P), which is communication-efficient since it needs to transmit a very small subsample from local machines to the central site, which is empirically feasible, together with the gradient of the loss. Practically, due to the use of Newton–Raphson iteration, the Hessian matrix can be computed more robustly using the subsampled data than using one local dataset. We also show that the DLPRE-P estimator is statistically efficient as the global estimator, which is based on putting all the datasets together. Furthermore, we propose a distributed regularized LPRE estimator (DRLPRE-P) to consider the variable selection problem in high dimension. A distributed algorithm based on the alternating direction method of multipliers (ADMM) is developed for implementing the DRLPRE-P. The oracle property holds for DRLPRE-P. Finally, simulation experiments and two real-world data analyses are conducted to illustrate the performance of our methods.