A Learning-based Distributed Algorithm for Personalized Aggregative Optimization

This paper addresses distributed aggregative optimization, i.e., a recently emerged framework in which agents in a network want to minimize the sum of local objective functions depending both on a local decision variable and on an aggregated version of all the variables (e.g., the mean). We consider a "personalized"set-up in which each local function is given by the sum of a known term and an unknown one capturing the user's dissatisfaction. We propose a novel data-driven distributed algorithm taking advantage of users' noisy feedback to learn the parameters of the unknown function concurrently with the optimization steps. We prove an upper bound for the dynamic regret related to (i) the initial conditions, (ii) the temporal variations of the objective functions, and (iii) the learning errors. Moreover, by considering the average dynamic regret, we prove that both initial conditions and learning errors do not affect the asymptotic performance of the algorithm. Finally, a numerical simulation in the context of opinion dynamics validates our findings.