3,614 research outputs found
Accelerating Consensus by Spectral Clustering and Polynomial Filters
It is known that polynomial filtering can accelerate the convergence towards
average consensus on an undirected network. In this paper the gain of a
second-order filtering is investigated. A set of graphs is determined for which
consensus can be attained in finite time, and a preconditioner is proposed to
adapt the undirected weights of any given graph to achieve fastest convergence
with the polynomial filter. The corresponding cost function differs from the
traditional spectral gap, as it favors grouping the eigenvalues in two
clusters. A possible loss of robustness of the polynomial filter is also
highlighted
Accelerated Consensus via Min-Sum Splitting
We apply the Min-Sum message-passing protocol to solve the consensus problem
in distributed optimization. We show that while the ordinary Min-Sum algorithm
does not converge, a modified version of it known as Splitting yields
convergence to the problem solution. We prove that a proper choice of the
tuning parameters allows Min-Sum Splitting to yield subdiffusive accelerated
convergence rates, matching the rates obtained by shift-register methods. The
acceleration scheme embodied by Min-Sum Splitting for the consensus problem
bears similarities with lifted Markov chains techniques and with multi-step
first order methods in convex optimization
Distributed Weight Selection in Consensus Protocols by Schatten Norm Minimization
In average consensus protocols, nodes in a network perform an iterative
weighted average of their estimates and those of their neighbors. The protocol
converges to the average of initial estimates of all nodes found in the
network. The speed of convergence of average consensus protocols depends on the
weights selected on links (to neighbors). We address in this paper how to
select the weights in a given network in order to have a fast speed of
convergence for these protocols. We approximate the problem of optimal weight
selection by the minimization of the Schatten p-norm of a matrix with some
constraints related to the connectivity of the underlying network. We then
provide a totally distributed gradient method to solve the Schatten norm
optimization problem. By tuning the parameter p in our proposed minimization,
we can simply trade-off the quality of the solution (i.e. the speed of
convergence) for communication/computation requirements (in terms of number of
messages exchanged and volume of data processed). Simulation results show that
our approach provides very good performance already for values of p that only
needs limited information exchange. The weight optimization iterative procedure
can also run in parallel with the consensus protocol and form a joint
consensus-optimization procedure.Comment: N° RR-8078 (2012
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