Decentralized Gradient Methods with Time-varying Uncoordinated Stepsizes: Convergence Analysis and Privacy Design

Decentralized optimization enables a network of agents to cooperatively optimize an overall objective function without a central coordinator and is gaining increased attention in domains as diverse as control, sensor networks, data mining, and robotics. However, the information sharing among agents in decentralized optimization also discloses agents' information, which is undesirable or even unacceptable when involved data are sensitive. This paper proposes two gradient based decentralized optimization algorithms that can protect participating agents' privacy without compromising optimization accuracy or incurring heavy communication/computational overhead. This is in distinct difference from differential privacy based approaches which have to trade optimization accuracy for privacy, or encryption based approaches which incur heavy communication and computational overhead. Both algorithms leverage a judiciously designed mixing matrix and time-varying uncoordinated stepsizes to enable privacy, one using diminishing stepsizes while the other using non-diminishing stepsizes. In both algorithms, when interacting with any one of its neighbors, a participating agent only needs to share one message in each iteration to reach convergence to an exact optimal solution, which is in contrast to most gradient-tracking based algorithms requiring every agent to share two messages (an optimization variable and an auxiliary gradient-tracking variable) under non-diminishing stepsizes. Furthermore, both algorithms can guarantee the privacy of a participating agent even when all information shared by the agent are accessible to an adversary, a scenario in which most existing accuracy-maintaining privacy approaches will fail to protect privacy. Simulation results confirm the effectiveness of the proposed algorithms.