Decentralized optimization with non-identical sampling in presence of stragglers

TL;DR

This paper analyzes decentralized consensus optimization with non-identical data sampling and stragglers, comparing heuristic methods for combining worker outputs, and concludes that neither method is optimal but they can perform differently under approximate consensus.

Contribution

It provides a unified analysis of non-identical sampling and variable work in decentralized optimization, evaluating heuristic methods and their convergence properties.

Findings

The second weighting method outperforms the first under approximate consensus.

Convergence is proven under perfect consensus with iid straggler statistics.

Neither heuristic method is optimal, and an optimal method does not exist.

Abstract

We consider decentralized consensus optimization when workers sample data from non-identical distributions and perform variable amounts of work due to slow nodes known as stragglers. The problem of non-identical distributions and the problem of variable amount of work have been previously studied separately. In our work we analyze them together under a unified system model. We study the convergence of the optimization algorithm when combining worker outputs under two heuristic methods: (1) weighting equally, and (2) weighting by the amount of work completed by each. We prove convergence of the two methods under perfect consensus, assuming straggler statistics are independent and identical across all workers for all iterations. Our numerical results show that under approximate consensus the second method outperforms the first method for both convex and non-convex objective functions. We make use of the theory on minimum variance unbiased estimator (MVUE) to evaluate the existence of an optimal method for combining worker outputs. While we conclude that neither of the two heuristic methods are optimal, we also show that an optimal method does not exist.