Paving the Way for Consensus: Convergence of Block Gossip Algorithms

Gossip protocols are popular methods for average consensus problems in distributed computing. We prove new convergence guarantees for a variety of such protocols, including path, clique, and synchronous pairwise gossip. These arise by exploiting the connection between these protocols and the block randomized Kaczmarz method for solving linear systems. Moreover, we extend existing convergence results for block randomized Kaczmarz to allow for a more general choice of blocks, rank-deficient systems, and provide a tighter convergence rate guarantee. Previous results for block randomized Kaczmarz assumed blocks formed a paving of the systems; our analysis, contrary to the title of our paper, generalizes to a broader class of blocks that we call coverings. We furthermore apply this analysis to inconsistent consensus models and obtain similar guarantees. An extensive empirical analysis of these methods is provided for a variety of synthetic and real networks.