  Distributed minimax estimation and distributed adaptive estimation under
communication constraints for Gaussian sequence model and white noise model are
studied. The minimax rate of convergence for distributed estimation over a
given Besov class, which serves as a benchmark for the cost of adaptation, is
established. We then quantify the exact communication cost for adaptation and
construct an optimally adaptive procedure for distributed estimation over a
range of Besov classes. The results demonstrate significant differences between
nonparametric function estimation in the distributed setting and the
conventional centralized setting. For global estimation, adaptation in general
cannot be achieved for free in the distributed setting. The new technical tools
to obtain the exact characterization for the cost of adaptation can be of
independent interest.
