Accelerated Consensus with Linear Rate of Convergence

We are concerned with the problem of achieving average consensus on fixed, undirected and connected graphs with an unknown graph topology. Motivated by Nesterov's accelerated gradient algorithm [1] [2], we describe a completely distributed consensus protocol that has a rate of convergence of O(q^2k/k²), where k is the iteration number and is dependent on the second largest eigenvalue modulus of weighting matrix constructed on the graph. By casting the protocol into a linear time-variant system, we show a dimension-free worst case convergence rate bound, that is, a rate independent of the scale of the network. We also discuss several convergence rate bounds for different choices of the weighting matrix in the protocol.