Re-weighting of Vector-weighted Mechanisms for Utility Maximization under Differential Privacy

We present practical aspects of implementing a pseudo posterior synthesizer for microdata dissemination under a new re-weighting strategy for utility maximization of released synthetic data. Our re-weighting strategy applies to any vector-weighting approach under which a vector of observation-indexed weight are used to downweight likelihood contributions for high disclosure risk records. We demonstrate our method on two different vector-weighted schemes that target high-risk records by exponentiating each of their likelihood contributions with a record-indexed weight, $\alpha_i \in [0,1]$ for record $i \in (1,\ldots,n)$. We compute the overall Lipschitz bound, $\Delta_{\boldsymbol{\alpha},\mathbf{x}}$, for the database $\mathbf{x}$, under each vector-weighted scheme where a local $\epsilon_{x} = 2\Delta_{\boldsymbol{\alpha},\mathbf{x}}$. Our new method for constructing record-indexed downeighting maximizes the data utility under any privacy budget for the vector-weighted synthesizers by adjusting the by-record weights, $(\alpha_{i})_{i = 1}^{n}$, such that their individual Lipschitz bounds, $\Delta_{\boldsymbol{\alpha},x_{i}}$, approach the bound for the entire database, $\Delta_{\boldsymbol{\alpha},\mathbf{x}}$. Our method asymptotically (as sample size grows) achieves an $(\epsilon = 2 \Delta_{\boldsymbol{\alpha}})-$ differential privacy (DP) guarantee, globally, over the space of databases, $\mathbf{x}\in\mathcal{X}$. We illustrate our methods using simulated count data with and without over-dispersion-induced skewness and compare the results to a scalar-weighted synthesizer under the Exponential Mechanism. We demonstrate our asymptotic DP result in a simulation study. We apply our methods to a sample of the Survey of Doctorate Recipients.