Distributed Mean Estimation with Limited Communication

Motivated by the need for distributed optimization algorithms with low communication cost, we study communication efficient algorithms to perform distributed mean estimation. We study scenarios in which each client sends one bit per dimension. We first show that for $d$ dimensional data with $n$ clients, a naive stochastic rounding approach yields a mean squared error $\Theta(d/n)$. We then show by applying a structured random rotation of the data (an $\mathcal{O}(d \log d)$ algorithm), the error can be reduced to $\mathcal{O}((\log d)/n)$. The algorithms and the analysis make no distributional assumptions on the data.