Randomized Distributed Mean Estimation: Accuracy vs Communication
This addresses a subproblem in distributed and federated optimization, offering incremental improvements in communication-efficient mean estimation.
The paper tackles the problem of estimating the average of vectors distributed across nodes under a communication budget, proposing a family of randomized algorithms that trade off communication cost and estimation error, achieving an error of O(r/n) when communicating one bit per coordinate.
We consider the problem of estimating the arithmetic average of a finite collection of real vectors stored in a distributed fashion across several compute nodes subject to a communication budget constraint. Our analysis does not rely on any statistical assumptions about the source of the vectors. This problem arises as a subproblem in many applications, including reduce-all operations within algorithms for distributed and federated optimization and learning. We propose a flexible family of randomized algorithms exploring the trade-off between expected communication cost and estimation error. Our family contains the full-communication and zero-error method on one extreme, and an $ε$-bit communication and ${\cal O}\left(1/(εn)\right)$ error method on the opposite extreme. In the special case where we communicate, in expectation, a single bit per coordinate of each vector, we improve upon existing results by obtaining $\mathcal{O}(r/n)$ error, where $r$ is the number of bits used to represent a floating point value.