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This paper investigates the problem of distributed stochastic approximationin multi-agent systems. The algorithm under study consists of two steps: alocal stochastic approximation step and a diffusion step which drives thenetwork to a consensus. The diffusion step uses row-stochastic matrices toweight the network exchanges. As opposed to previous works, exchange matricesare not supposed to be doubly stochastic, and may also depend on the pastestimate.  We prove that non-doubly stochastic matrices generally influence the limitpoints of the algorithm. Nevertheless, the limit points are not affected by thechoice of the matrices provided that the latter are doubly-stochastic inexpectation. This conclusion legitimates the use of broadcast-like diffusionprotocols, which are easier to implement. Next, by means of a central limittheorem, we prove that doubly stochastic protocols perform asymptotically aswell as centralized algorithms and we quantify the degradation caused by theuse of non doubly stochastic matrices. Throughout the paper, a special emphasisis put on the special case of distributed non-convex optimization as anillustration of our results.

Success and Failure of Adaptation-Diffusion Algorithms for Consensus in Multi-Agent Networks