Differentially Private Online Distributed Aggregative Games With Time-Varying and Non-Identical Communication and Feedback Delays

This paper investigates online distributed aggregative games with time-varying cost functions, where agents are interconnected through an unbalanced communication graph. Due to the distributed and noncooperative nature of the game, some curious agents may wish to steal sensitive information from neighboring agents during parameter exchanges. Additionally, communication delays arising from network congestion, particularly in wireless settings, as well as feedback delays, can hinder the convergence of agents to a Nash equilibrium. Although a recent work addressed both communication and feedback delays in aggregative games, it is based on the unrealistic assumption that the delays are fixed over time and identical across agents. Hence, the case of time-varying and non-identical delays across agents has never been considered in aggregative games. In this work, we address the combined challenges of privacy leakage with time-varying and non-identical communication and feedback delays for the first time. We propose an online distributed dual averaging algorithm that simultaneously tackles these challenges while achieving a provably low regret bound. Our simulation result shows that the running average of each client’s local action converges over time.

💡 Research Summary

This paper tackles the complex problem of designing a distributed algorithm for online aggregative games that is robust to real-world network imperfections and privacy concerns. The research is set in a scenario where multiple non-cooperative agents, interconnected via a time-varying and unbalanced communication graph, aim to minimize their individual time-varying cost functions, which depend on an aggregate of all agents’ actions. The work simultaneously addresses three significant challenges that have not been combined in prior literature: privacy leakage during inter-agent communication, time-varying and non-identical communication delays, and time-varying feedback delays in observing cost functions.

The authors begin by formalizing the online distributed aggregative game, where the solution concept is a time-varying Nash equilibrium. They identify the limitations of existing works, which often assume fixed or identical delays, and highlight the lack of a unified approach that incorporates differential privacy with such heterogeneous, time-varying delays. To bridge this gap, the paper proposes a novel Online Distributed Dual Averaging algorithm with Delays and Privacy (ODDA-DP).

The core algorithm operates in a fully decentralized manner. To preserve privacy, each agent injects calibrated Laplace noise into its action estimate before sharing it with neighbors. To handle delays, the algorithm’s update steps strategically utilize outdated information: agents use delayed gradients (computed from cost functions observed with a lag) and delayed action estimates received from neighbors (which suffered communication delays). Furthermore, to counteract the asymmetry of the unbalanced graph, the algorithm incorporates an estimate of the left eigenvector corresponding to the eigenvalue of 1 for the row-stochastic weight matrix, ensuring consensus convergence.

A major theoretical contribution is the rigorous analysis of the algorithm’s performance. The authors prove that despite the added noise and the use of delayed information, the algorithm achieves a sublinear dynamic regret bound of O(√T). This bound depends on the maximum bounds for both communication and feedback delays and the connectivity of the graph. The analysis employs a technique of augmenting the original network graph with virtual nodes to model the delayed information flows, simplifying the convergence proof. Additionally, the paper formally demonstrates that the Laplace noise mechanism provides (ε, δ)-differential privacy for each agent’s sensitive action sequence.

Finally, the theoretical claims are supported by numerical simulations. The results visually demonstrate that the running average of each agent’s local action converges over time, validating the algorithm’s ability to drive agents towards a consensus that approximates the time-varying Nash equilibrium, even under the combined adversities of privacy noise and heterogeneous, time-varying delays. In conclusion, this work provides a comprehensive and robust solution for privacy-preserving, delay-tolerant distributed optimization in dynamic game-theoretic environments.