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

arXiv: https://arxiv.org/abs/2107.00179 · HTML: https://mlsanity.com/paper/2107.00179/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 39 references that are in the map.
- First posted on arXiv 1 Jul 2021.

## Builds on (most cited first, 9 of 9)

- [Distributed Gaussian Processes](https://mlsanity.com/paper/1502.02843/) (2015)
- [Communication Lower Bounds for Statistical Estimation Problems via a Distributed Data Processing Inequality](https://mlsanity.com/paper/1506.07216/) (2015)
- [Geometric Lower Bounds for Distributed Parameter Estimation under Communication Constraints](https://mlsanity.com/paper/1802.08417/) (2018)
- [On Communication Cost of Distributed Statistical Estimation and Dimensionality](https://mlsanity.com/paper/1405.1665/) (2014)
- [Distributed Gaussian Mean Estimation under Communication Constraints: Optimal Rates and Communication-Efficient Algorithms](https://mlsanity.com/paper/2001.08877/) (2020)
- [Adaptive distributed methods under communication constraints](https://mlsanity.com/paper/1804.00864/) (2018)
- [Distributed Nonparametric Regression under Communication Constraints](https://mlsanity.com/paper/1803.01302/) (2018)
- [Distributed function estimation: adaptation using minimal communication](https://mlsanity.com/paper/2003.12838/) (2020)
- [Lower Bounds for Learning Distributions under Communication Constraints via Fisher Information](https://mlsanity.com/paper/1902.02890/) (2019)

## Built on by (most cited first, 4 of 4)

- [Order-Optimal Sequential 1-Bit Mean Estimation in General Tail Regimes](https://mlsanity.com/paper/2604.07796/) (2026)
- [Sequential 1-bit Mean Estimation with Near-Optimal Sample Complexity](https://mlsanity.com/paper/2509.21940/) (2025)
- [Optimal high-dimensional and nonparametric distributed testing under communication constraints](https://mlsanity.com/paper/2202.00968/) (2022)
- [Distributed Nonparametric Estimation under Communication Constraints](https://mlsanity.com/paper/2204.10373/) (2022)

## Abstract (arXiv)

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.