Random pairwise gossip on CAT(0) metric spaces

In the context of sensor networks, gossip algorithms are a popular, well established technique for achieving consensus when sensor data is encoded in Euclidean spaces. The algorithm also has several extensions to nonlinear data spaces. Most of these extensions deal with Riemannian data spaces and use gradient descent techniques to derive a generalization of the gossip algorithm. However, their heavy reliance on calculus tools hides the simple metric properties that make gossip algorithms work. This paper studies the pairwise gossip algorithm from a purely metric perspective. Focusing on the essential property that makes pairwise midpoint gossip converge on Riemannian manifolds, this paper argues that the notion of CAT(0) space is perfectly suited to the analysis of gossip algorithms. The transition from a Riemannian to a purely metrical framework has three virtues: it simplifies the proofs, provably preserves convergence speed of pairwise gossip, yet in a more general framework, and paves the way for new applications, as illustrated by our numerical experiments on robots arms viewed as points in the metric graph associated with the free group.