Almost sure convergence of consensus algorithms by relaxed projection mappings

In this contribution we present a stronger notion of almost sure convergence for a large class of consensus algorithms including also asynchronous updates. We introduce the concept of the so-called relaxed projection algorithms and show that many consensus algorithms can be interpreted as such relaxed projection updates. It is well known that such algorithms converge to a solution lying in the intersection of the projections. The convergence of such algorithms is, however, guaranteed only for deterministic ordering of the projections. Since we are interested in random data exchanges, we analyze the convergence in case of random orderings of the projections and show that the algorithms converge in the underrelaxed case even for time-varying and individual mixing parameters.