Differentially Private Parameter Estimation: Optimal Noise Insertion and Data Owner Selection

In this paper, we study differentially private parameter estimation in which a data acquisitor (DA) accesses data (or signals) from multiple privacy-aware data owners (DOs) to estimate some random parameters. To ensure differential privacy, the DOs add Laplace noises to their private signals and only reveal the noisy signals to the DA. Our goal is to add optimal amount of noises (measured by their variances) so that the mean squared error (MSE) of the DA's estimate is minimized while differential privacy is satisfied. In the general case, the optimal private estimation can be formulated as a semidefinite program (SDP), which can be readily solved by off-the-shelf optimization methods. In the special case where different DOs have uncorrelated signals, the optimization problem is decomposed across DOs and can be solved very efficiently in almost closed-form. We observe that, in the optimal solution, the DOs should add more noises to the signal entries that are less useful for estimation. Further, when the DA has DO selection constraint (e.g., due to limited budget), a relaxed SDP is put forth to compute a suboptimal solution. Finally, several numerical examples are presented.