Trade-Off Between Privacy and Accuracy in Resilient Vector Consensus

This paper studies privacy-preserving resilient vector consensus in multi-agent systems where the number of tolerant faulty agents is limited. The objective is to guarantee that state vectors of all normal agents converge to a common state vector in the convex hull formed by their initial state vectors privately. To address this issue, we consider a modification of an existing algorithm called Approximate Distributed Robust Convergence Using Centerpoints (ADRC), named Differentially Private ADRC (DP-ADRC), where each agent adds noise to its state at each iteration. We first show the convergence and c-differential privacy of DP-ADRC. Then, we characterize the change of the convex hull caused by the noise from two perspectives, i.e., the distance between the convex hulls with and without noises and the area change of the convex hull under a special case, respectively. Specifically, we use Chebyshev's inequality to characterize the upper bound of the Hausdorff distance between the two convex hulls given a probability, where only partial dimensions of each state are added with noises. To characterize the area change, we consider adding noise to protect initial states during the first iteration and obtain the probability density function of the area difference between two convex hulls in the two-dimensional case. Finally, we perform numerical simulations to demonstrate the theoretical results.