# Distributed Gaussian Mean Estimation under Communication Constraints: Optimal Rates and Communication-Efficient Algorithms

arXiv: https://arxiv.org/abs/2001.08877 · HTML: https://mlsanity.com/paper/2001.08877/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2020.

## Built on by (most cited first, 17 of 17)

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- [Robust Mean Estimation under Quantization](https://mlsanity.com/paper/2601.07074/) (2026)
- [Distributed Nonparametric Function Estimation: Optimal Rate of Convergence and Cost of Adaptation](https://mlsanity.com/paper/2107.00179/) (2021)
- [Distributed Sparse Normal Means Estimation with Sublinear Communication](https://mlsanity.com/paper/2102.03060/) (2021)
- [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)
- [Open Problem: Is Interaction Necessary for Order-Optimal 1-bit Mean Estimation?](https://mlsanity.com/paper/2607.02896/) (2026)
- [Interaction Is Not Necessary for Order-Optimal 1-Bit Mean Estimation](https://mlsanity.com/paper/2608.02538/) (2026)
- [Stock Trading Optimization through Model-based Reinforcement Learning with Resistance Support Relative Strength](https://mlsanity.com/paper/2205.15056/) (2022)
- [Data-light Uncertainty Set Merging with Admissibility](https://mlsanity.com/paper/2410.12201/) (2024)
- [Information-Theoretic Bounds for Sparse Covariance Estimation in the Vertical-Split Distributed Model](https://mlsanity.com/paper/2606.07124/) (2026)
- [Joint Extremum Compression and Detection of a Time-Delayed Signal for Distributed Sensing](https://mlsanity.com/paper/2607.10874/) (2026)
- [Exact Minimax One-Bit Unbiased Compression: Heavy-Tail Necessity and Finite-Randomness Approximation](https://mlsanity.com/paper/2609.27860/) (2026)

## Abstract (arXiv)

We study distributed estimation of a Gaussian mean under communication constraints in a decision theoretical framework. Minimax rates of convergence, which characterize the tradeoff between the communication costs and statistical accuracy, are established in both the univariate and multivariate settings. Communication-efficient and statistically optimal procedures are developed. In the univariate case, the optimal rate depends only on the total communication budget, so long as each local machine has at least one bit. However, in the multivariate case, the minimax rate depends on the specific allocations of the communication budgets among the local machines. Although optimal estimation of a Gaussian mean is relatively simple in the conventional setting, it is quite involved under the communication constraints, both in terms of the optimal procedure design and lower bound argument. The techniques developed in this paper can be of independent interest. An essential step is the decomposition of the minimax estimation problem into two stages, localization and refinement. This critical decomposition provides a framework for both the lower bound analysis and optimal procedure design.