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

TL;DR

This paper investigates the limits of distributed nonparametric function estimation, establishing optimal convergence rates and quantifying the communication costs necessary for adaptation across Besov classes, highlighting key differences from centralized estimation.

Contribution

It provides the first characterization of the optimal convergence rate and the exact communication cost for adaptation in distributed nonparametric estimation over Besov classes.

Findings

Established the minimax rate of convergence for distributed estimation.

Quantified the exact communication cost for adaptation.

Constructed an optimally adaptive estimation procedure.

Abstract

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.