Distributed estimation of the left Perron eigenvector of non-column stochastic protocols for distributed stochastic approximation

The present paper addresses the problem of distributed stochastic approximation in multi-agent systems in a particular case. The algorithm under study consists of two steps: a local stochastic approximation step and a diffusion step which drives the network to a consensus. We focus on the case when only row-stochastic matrices are considered in the diffusion step to weight the network exchanges. Thus, contrarily to previous works, exchange matrices are not supposed to be doubly stochastic. Recent results show that the limit points are influenced by the left Perron eigenvector of the expectation matrix representing the communication model. Based on these results, we propose a distributed and asynchronous algorithm to compute this vector on a previous stage. By including the knowledge in the local stochastic step we lead the network converge towards the sought consensus. We illustrate the performance of the proposed algorithm when considering the problem of distributed optimization.