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Distributed minimax estimation and distributed adaptive estimation undercommunication constraints for Gaussian sequence model and white noise model arestudied. The minimax rate of convergence for distributed estimation over agiven Besov class, which serves as a benchmark for the cost of adaptation, isestablished. We then quantify the exact communication cost for adaptation andconstruct an optimally adaptive procedure for distributed estimation over arange of Besov classes. The results demonstrate significant differences betweennonparametric function estimation in the distributed setting and theconventional centralized setting. For global estimation, adaptation in generalcannot be achieved for free in the distributed setting. The new technical toolsto obtain the exact characterization for the cost of adaptation can be ofindependent interest.

Distributed Nonparametric Function Estimation: Optimal Rate of Convergence and Cost of Adaptation