Accurate performance analysis of distributed adaptive differential privacy estimation

Abstract

The incorporation of differential privacy mechanisms within the distributed parameter estimation problem has attracted significant attention. However, existing differential privacy algorithms predominantly focus on deriving upper bounds on estimation error, which lack precise quantification of error magnitudes. In this paper, we propose a distributed privacy-preserving least mean squares (LMS) algorithm with a noise injection mechanism for estimating unknown time-varying parameters in stochastic regression models. To derive the accurate performance of the algorithm, we first establish the theoretical upper bound for the estimation error under the cooperative excitation condition, which requires neither independence nor stationarity of the regression vectors. The mean square estimation error matrix is then approximated through a linear deterministic difference matrix equation, rigorously quantifying the relationship between noise injection and estimation accuracy. Finally, a simulation example is provided to verify the effectiveness of the proposed algorithm.