Wyner-Ziv Estimators for Distributed Mean Estimation with Side Information and Optimization

TL;DR

This paper introduces Wyner-Ziv estimators for distributed mean estimation that leverage side information to achieve near-optimal communication efficiency, with applications in federated learning and decentralized optimization.

Contribution

The paper proposes novel Wyner-Ziv estimators for distributed mean estimation that are efficient, near-optimal, and adaptable to scenarios with or without prior knowledge of data-side information distance.

Findings

Efficient Wyner-Ziv estimators are near-optimal when data-distance bounds are known.

Universal Wyner-Ziv estimator with correlated sampling offers practical benefits without prior data knowledge.

Algorithms using these estimators achieve near-optimal convergence in distributed optimization.

Abstract

Communication efficient distributed mean estimation is an important primitive that arises in many distributed learning and optimization scenarios such as federated learning. Without any probabilistic assumptions on the underlying data, we study the problem of distributed mean estimation where the server has access to side information. We propose \emph{Wyner-Ziv estimators}, which are communication and computationally efficient and near-optimal when an upper bound for the distance between the side information and the data is known. As a corollary, we also show that our algorithms provide efficient schemes for the classic Wyner-Ziv problem in information theory. In a different direction, when there is no knowledge assumed about the distance between side information and the data, we present an alternative Wyner-Ziv estimator that uses correlated sampling. This latter setting offers {\em universal recovery guarantees}, and perhaps will be of interest in practice when the number of users is large and keeping track of the distances between the data and the side information may not be possible. With this mean estimator at our disposal, we revisit basic problems in decentralized optimization and compression where our Wyner-Ziv estimator yields algorithms with almost optimal performance. First, we consider the problem of communication constrained distributed optimization and provide an algorithm which attains the optimal convergence rate by exploiting the fact that the gradient estimates are close to each other. Specifically, the gradient compression scheme in our algorithm first uses half of the parties to form side information and then uses our Wyner-Ziv estimator to compress the remaining half of the gradient estimates.