Differentially Private Accelerated Distributed Algorithm for Aggregative Optimization

This article studies the distributed aggregative optimization (DAO) problem, wherein each agent's local objective function depends not only on its own decision variables but also on an aggregate term involving all agents' decisions. In such settings, frequent information exchange among agents raises serious privacy concerns, as sensitive information may be inferred from shared data. To address this issue, we propose a differentially private accelerated distributed gradient tracking algorithm that integrates techniques from distributed dynamic average consensus, the heavy-ball momentum method, and differential privacy (DP). Specifically, to preserve privacy, the exchanged information is perturbed with independent Laplace noise. Moreover, our algorithm uses a noise deduction mechanism to prevent the accumulation of errors caused by noise during the estimation of aggregate variables and local gradients, thereby ensuring the algorithm's accuracy. Under the assumption that the global objective function is strongly convex and has Lipschitz-continuous gradients, we rigorously prove that the proposed algorithm achieves linear convergence in the mean-square error sense. In addition, we derive explicit suboptimality bounds and formally establish that the algorithm satisfies $\epsilon $ -DP. Finally, numerical simulations are provided to validate the effectiveness of the proposed method.