Distributed Mean Estimation with Limited Communication

TL;DR

This paper introduces communication-efficient algorithms for distributed mean estimation that do not rely on probabilistic data assumptions, achieving near-optimal error rates with minimal communication, and demonstrates their effectiveness in distributed learning tasks.

Contribution

The paper proposes novel, non-probabilistic algorithms for distributed mean estimation that significantly reduce error and communication costs, with proven optimality in the minimax sense.

Findings

Structured random rotation reduces error to O((log d)/n)

Advanced coding strategy achieves O(1/n) error with constant bits per dimension

Algorithms are effective in distributed Lloyd's for k-means and PCA

Abstract

Motivated by the need for distributed learning and optimization algorithms with low communication cost, we study communication efficient algorithms for distributed mean estimation. Unlike previous works, we make no probabilistic assumptions on the data. We first show that for $d$ dimensional data with $n$ clients, a naive stochastic binary rounding approach yields a mean squared error (MSE) of $\Theta(d/n)$ and uses a constant number of bits per dimension per client. We then extend this naive algorithm in two ways: we show that applying a structured random rotation before quantization reduces the error to $\mathcal{O}((\log d)/n)$ and a better coding strategy further reduces the error to $\mathcal{O}(1/n)$ and uses a constant number of bits per dimension per client. We also show that the latter coding strategy is optimal up to a constant in the minimax sense i.e., it achieves the best MSE for a given communication cost. We finally demonstrate the practicality of our algorithms by applying them to distributed Lloyd's algorithm for k-means and power iteration for PCA.