  This work carries out a detailed transient analysis of the learning behavior
of multi-agent networks, and reveals interesting results about the learning
abilities of distributed strategies. Among other results, the analysis reveals
how combination policies influence the learning process of networked agents,
and how these policies can steer the convergence point towards any of many
possible Pareto optimal solutions. The results also establish that the learning
process of an adaptive network undergoes three (rather than two) well-defined
stages of evolution with distinctive convergence rates during the first two
stages, while attaining a finite mean-square-error (MSE) level in the last
stage. The analysis reveals what aspects of the network topology influence
performance directly and suggests design procedures that can optimize
performance by adjusting the relevant topology parameters. Interestingly, it is
further shown that, in the adaptation regime, each agent in a sparsely
connected network is able to achieve the same performance level as that of a
centralized stochastic-gradient strategy even for left-stochastic combination
strategies. These results lead to a deeper understanding and useful insights on
the convergence behavior of coupled distributed learners. The results also lead
to effective design mechanisms to help diffuse information more thoroughly over
networks.
