  We consider distributed multitask learning problems over a network of agents
where each agent is interested in estimating its own parameter vector, also
called task, and where the tasks at neighboring agents are related according to
a set of linear equality constraints. Each agent possesses its own convex cost
function of its parameter vector and a set of linear equality constraints
involving its own parameter vector and the parameter vectors of its neighboring
agents. We propose an adaptive stochastic algorithm based on the projection
gradient method and diffusion strategies in order to allow the network to
optimize the individual costs subject to all constraints. Although the
derivation is carried out for linear equality constraints, the technique can be
applied to other forms of convex constraints. We conduct a detailed
mean-square-error analysis of the proposed algorithm and derive closed-form
expressions to predict its learning behavior. We provide simulations to
illustrate the theoretical findings. Finally, the algorithm is employed for
solving two problems in a distributed manner: a minimum-cost flow problem over
a network and a space-time varying field reconstruction problem.
