  The diffusion LMS algorithm has been extensively studied in recent years.
This efficient strategy allows to address distributed optimization problems
over networks in the case where nodes have to collaboratively estimate a single
parameter vector. Problems of this type are referred to as single-task
problems. Nevertheless, there are several problems in practice that are
multitask-oriented in the sense that the optimum parameter vector may not be
the same for every node. This brings up the issue of studying the performance
of the diffusion LMS algorithm when it is run, either intentionally or
unintentionally, in a multitask environment. In this paper, we conduct a
theoretical analysis on the stochastic behavior of diffusion LMS in the case
where the so-called single-task hypothesis is violated. We explain under what
conditions diffusion LMS continues to deliver performance superior to
non-cooperative strategies in the multitask environment. When the conditions
are violated, we explain how to endow the nodes with the ability to cluster
with other similar nodes to remove bias. We propose an unsupervised clustering
strategy that allows each node to select, via adaptive adjustments of
combination weights, the neighboring nodes with which it can collaborate to
estimate a common parameter vector. Simulations are presented to illustrate the
theoretical results, and to demonstrate the efficiency of the proposed
clustering strategy. The framework is applied to a useful problem involving a
multi-target tracking task.
