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We propose a strategy for computing the isotonic least-squares estimate of amonotone function in a general regression setting where the data aredistributed across different servers and the observations across servers,though independent, can come from heterogeneous sub-populations, therebyviolating the identically distributed assumption. Our strategy fixes thesuper-efficiency phenomenon observed in prior work on distributed computing inthe isotonic regression framework, where averaging several isotonic estimates(each computed at a local server) on a central server produces super-efficientestimates that do not replicate the properties of the global isotonicestimator, i.e. the isotonic estimate that would be constructed by transferringall the data to a single server. The new estimator proposed in this paper worksby smoothing the data on each local server, communicating the smoothedsummaries to the central server, and then computing an isotonic estimate at thecentral server, and is shown to replicate the asymptotic properties of theglobal estimator, and also overcome the super-efficiency phenomenon exhibitedby earlier estimators. For data on $N$ observations, the new estimator can beconstructed by transferring data just over order $N^{1/3}$ across servers [ascompared to transferring data of order $N$ to compute the global isotonicestimator], and requires the same order of computing time as the globalestimator.

Removing the Curse of Superefficiency: an Effective Strategy For Distributed Computing in Isotonic Regression