Success and Failure of Adaptation-Diffusion Algorithms for Consensus in Multi-Agent Networks

TL;DR

This paper analyzes the impact of stochastic diffusion matrices on consensus algorithms in multi-agent systems, showing that doubly stochastic matrices in expectation preserve limit points and asymptotic performance, facilitating easier broadcast protocols.

Contribution

It demonstrates that non-doubly stochastic matrices affect limit points unless they are doubly stochastic in expectation, and provides a central limit theorem for performance comparison.

Findings

Doubly stochastic matrices in expectation preserve limit points.

Asymptotic performance of doubly stochastic protocols matches centralized algorithms.

Non doubly stochastic matrices cause degradation in performance.

Abstract

This paper investigates the problem of distributed stochastic approximation in multi-agent systems. The algorithm under study consists of two steps: a local stochastic approximation step and a diffusion step which drives the network to a consensus. The diffusion step uses row-stochastic matrices to weight the network exchanges. As opposed to previous works, exchange matrices are not supposed to be doubly stochastic, and may also depend on the past estimate. We prove that non-doubly stochastic matrices generally influence the limit points of the algorithm. Nevertheless, the limit points are not affected by the choice of the matrices provided that the latter are doubly-stochastic in expectation. This conclusion legitimates the use of broadcast-like diffusion protocols, which are easier to implement. Next, by means of a central limit theorem, we prove that doubly stochastic protocols perform asymptotically as well as centralized algorithms and we quantify the degradation caused by the use of non doubly stochastic matrices. Throughout the paper, a special emphasis is put on the special case of distributed non-convex optimization as an illustration of our results.