On the cost of differential privacy in distributed control systems

Individuals sharing information can improve the cost or performance of a distributed control system. But, sharing may also violate privacy. We develop a general framework for studying the cost of differential privacy in systems where a collection of agents, with coupled dynamics, communicate for sensing their shared environment while pursuing individual preferences. First, we propose a communication strategy that relies on adding carefully chosen random noise to agent states and show that it preserves differential privacy. Of course, the higher the standard deviation of the noise, the higher the cost of privacy. For linear distributed control systems with quadratic cost functions, the standard deviation becomes independent of the number agents and it decays with the maximum eigenvalue of the dynamics matrix. Furthermore, for stable dynamics, the noise to be added is independent of the number of agents as well as the time horizon up to which privacy is desired. Finally, we show that the cost of ε-differential privacy up to time T, for a linear stable system with N agents, is upper bounded by O(T³/Nε² ).