On the benefits of diffusion cooperation for distributed optimization and learning

This work characterizes the nature of the limit point of distributed strategies for adaptation and learning over networks in the general case when the combination policy is not necessarily doubly stochastic and when the individual risks do not necessarily share a common minimizer. It is shown that, for sufficiently small step-sizes, the limiting behavior of the network is mainly influenced by the right-eigenvector of the combination policy corresponding to the single eigenvalue at one. It is also shown that the limit point of the network is the unique solution to a certain fixed-point equation determined by the entries of this eigenvector. The arguments show further that even when only partial information is available to the agents, cooperation over a connected network enables the agents to attain the same level of performance as a centralized solution.