Order Optimal One-Shot Distributed Learning

We consider distributed statistical optimization in one-shot setting, where there are m machines each observing n i.i.d. samples. Based on its observed samples, each machine then sends an O(log(mn))-length message to a server, at which a parameter minimizing an expected loss is to be estimated. We propose an algorithm called Multi-Resolution Estimator (MRE) whose expected error is no larger than tildeObig(m^-1/max(d,2) n-1/2big), where d is the dimension of the parameter space. This error bound meets existing lower bounds up to poly-logarithmic factors, and is thereby order optimal. The expected error of MRE, unlike existing algorithms, tends to zero as the number of machines (m) goes to infinity, even when the number of samples per machine (n) remains upper bounded by a constant. This property of the MRE algorithm makes it applicable in new machine learning paradigms where m is much larger than n.