Data-Driven Distributed Optimization via Aggregative Tracking and Deep Learning

In this paper, we propose a novel distributed data-driven optimization scheme. In particular, we focus on the so-called aggregative framework, namely, the scenario in which a set of agents aim to cooperatively minimize the sum of local costs, each depending on both local decision variables and an aggregation of all of them. We consider a data-driven setup in which each objective function is unknown and can be only sampled at a single point per iteration (thanks to, e.g., feedback from human users or physical sensors). We address this scenario through a distributed algorithm that combines three key components: (i) a learning part that leverages neural networks to learn the local cost functions descent direction, (ii) an optimization routine that steers the estimates according to the learned direction to minimize the global cost, and (iii) a tracking mechanism that locally reconstructs the unavailable global quantities. By using tools from system theory, i.e., timescale separation and averaging theory, we formally prove that, in strongly convex setups, the overall distributed strategy linearly converges in a neighborhood of the optimal solution whose radius depends on the given accuracy capabilities of the neural networks. Finally, we corroborate the theoretical results with numerical simulations.