  In the era of big data, it is necessary to split extremely large data sets
across multiple computing nodes and construct estimators using the distributed
data. When designing distributed estimators, it is desirable to minimize the
amount of communication across the network because transmission between
computers is slow in comparison to computations in a single computer. Our work
provides a general framework for understanding the behavior of distributed
estimation under communication constraints for nonparametric problems. We
provide results for a broad class of models, moving beyond the Gaussian
framework that dominates the literature. As concrete examples we derive minimax
lower and matching upper bounds in the distributed regression, density
estimation, classification, Poisson regression and volatility estimation models
under communication constraints. To assist with this, we provide sufficient
conditions that can be easily verified in all of our examples.
