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