Differentially private projected gradient tracking for distributed constrained optimization

This paper investigates a distributed constrained optimization problem with privacy concerns over a multi-agent network, where each agent holds a private objective function and seeks a global optimizer subject to a global closed convex set of constraints in a distributed and privacy-preserving manner. To address this problem, a differentially private projected gradient tracking algorithm is proposed. The main idea is to introduce indirect projection in gradient tracking to handle the global constraints and to decompose the gradient tracking state into two sub-states, with the shared sub-state being perturbed by decaying Laplace noise, to establish differential privacy (DP). Two time scales and a lazy update rule are employed to facilitate the convergence analysis. It is demonstrated that the proposed algorithm simultaneously preserves linear convergence rates and ϵ-DP without requiring bounded gradients. Finally, numerical simulations are presented to verify the theoretical results. © 2026