Distributed subgradient method under random communication topology – the effect of the probability distribution of the random graph on the performance metrics

In this note we study the performance metrics (rate of convergence and guaranteed region of convergence) of a multi-agent subgradient method for optimizing a sum of convex functions. We assume that the agents exchange information according to a communication topology modeled as a random graph, independent of other time instances. Under a strong convexity type of assumption, we express the performance metrics directly as functions of the estimates of the optimal decision vector. We emphasize how the probability distribution of the random graph affects the upper bounds on the performance metrics. This provide a guide for tuning the parameters of the communication protocol such that good performance of the multi-agent subgradient method is ensured. We perform the tuning of the protocol parameters for two communication scenarios. In the first scenario, we assume a randomized scheme for link activation with no-error transmissions while in the second scenario we use a pre-established order of transmissions but we consider the interference effect. Both these scenarios are applied on a small world type of topology.