A distributed one-step estimator

Distributed statistical inference has recently attracted enormous attention. Many existing work focuses on the averaging estimator, e.g., Zhang and Duchi (J Mach Learn Res 14:3321–3363, 2013) together with many others. We propose a one-step approach to enhance a simple-averaging based distributed estimator by utilizing a single Newton–Raphson updating. We derive the corresponding asymptotic properties of the newly proposed estimator. We find that the proposed one-step estimator enjoys the same asymptotic properties as the idealized centralized estimator. In particular, the asymptotic normality was established for the proposed estimator, while other competitors may not enjoy the same property. The proposed one-step approach merely requires one additional round of communication in relative to the averaging estimator; so the extra communication burden is insignificant. The proposed one-step approach leads to a lower upper bound of the mean squared error than other alternatives. In finite sample cases, numerical examples show that the proposed estimator outperforms the simple averaging estimator with a large margin in terms of the sample mean squared error. A potential application of the one-step approach is that one can use multiple machines to speed up large scale statistical inference with little compromise in the quality of estimators. The proposed method becomes more valuable when data can only be available at distributed machines with limited communication bandwidth.