Performance Evaluation of the Consensus-Based Distributed Subgradient Method Under Random Communication Topologies

• Published in: IEEE journal of selected topics in signal processing Vol. 5; no. 4; pp. 754 - 771
• Main Authors: Matei, Ion, Baras, John S.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-08-2011; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Convex analysis; Convex functions; Decisions; Distributed; Estimates; Mathematical analysis; Mathematical models; Measurement; Optimization; Probability distribution; Protocols; stochastic systems; Studies; sub-gradient methods; system identification; Topology; Upper bound; Upper bounds; Vectors (mathematics)
• ISSN: 1932-4553; 1941-0484
• DOI: 10.1109/JSTSP.2011.2120593

We investigate collaborative optimization of an objective function expressed as a sum of local convex functions, when the agents make decisions in a distributed manner using local information, while the communication topology used to exchange messages and information is modeled by a graph-valued random process, assumed independent and identically distributed. Specifically, we study the performance of the consensus-based multi-agent distributed subgradient method and show how it depends on the probability distribution of the random graph. For the case of a constant stepsize, we first give an upper bound on the difference between the objective function, evaluated at the agents' estimates of the optimal decision vector, and the optimal value. Second, for a particular class of convex functions, we give an upper bound on the distances between the agents' estimates of the optimal decision vector and the minimizer. In addition, we provide the rate of convergence to zero of the time varying component of the aforementioned upper bound. The addressed metrics are evaluated via their expected values. As an application, we show how the distributed optimization algorithm can be used to perform collaborative system identification and provide numerical experiments under the randomized and broadcast gossip protocols.