Distributed Statistical Estimation and Rates of Convergence in Normal Approximation

This paper presents new algorithms for distributed statistical estimation that can take advantage of the divide-and-conquer approach. We show that one of the key benefits attained by an appropriate divide-and-conquer strategy is robustness, an important characteristic of large distributed systems. We introduce a class of algorithms that are based on the properties of the geometric median, establish connections between performance of these distributed algorithms and rates of convergence in normal approximation, and provide tight deviations guarantees for resulting estimators in the form of exponential concentration inequalities. We illustrate our techniques with several examples; in particular, we obtain new results for the median-of-means estimator, as well as provide performance guarantees for robust distributed maximum likelihood estimation.