Differential Privacy and Minimum-Variance Unbiased Estimation in Multi-agent Control Systems

In a discrete-time linear multi-agent control system, where the agents are coupled via an environmental state, knowledge of the environmental state is desirable to control the agents locally. However, since the environmental state depends on the behavior of the agents, sharing it directly among these agents jeopardizes the privacy of the agents' pro les, de ned as the combination of the agents' initial states and the sequence of local control inputs over time. A commonly used solution is to randomize the environmental state before sharing { this leads to a natural trade-o between the privacy of the agents' pro les and the variance of estimating the environmental state. By treating the multi-agent system as a probabilistic model of the environmental state parametrized by the agents' pro les, we show that when the agents' pro les is "-di erentially private, there is a lower bound on the `1 induced norm of the covariance matrix of the minimum-variance unbiased estimator of the environmental state. This lower bound is achieved by a randomized mechanism that uses Laplace noise.