A Deep Learning Approach for Distributed Aggregative Optimization with Users' Feedback.

We propose a novel distributed data-driven scheme for online aggregative optimization, i.e., the framework in which agents in a network aim to cooperatively minimize the sum of local time-varying costs, each depending on a local decision variable and an aggregation of all of them. We consider a "personalized" setup in which each cost exhibits a term capturing the user's dissatisfaction and, thus, is unknown. We enhance an existing distributed optimization scheme by endowing it with a learning mechanism based on neural networks that estimate the missing part of the gradient via users' feedback about the cost. Our algorithm combines two loops with different timescales devoted to performing optimization and learning steps. In turn, the proposed scheme also embeds a distributed consensus mechanism aimed at locally reconstructing the unavailable global information due to the presence of the aggregative variable. We prove an upper bound for the dynamic regret related to (i) the initial conditions, (ii) the temporal variations of the functions, and (iii) the learning errors about the unknown cost. Finally, we test our method via numerical simulations.