Privacy of Synthetic Data: A Statistical Framework

Privacy-preserving data analysis is emerging as a challenging problem with far-reaching impact. In particular, synthetic data are a promising concept toward solving the aporetic conflict between data privacy and data sharing. Yet, it is known that accurately generating private, synthetic data of certain kinds is NP-hard. We develop a statistical framework for differentially private synthetic data, which enables us to circumvent the computational hardness of the problem. We consider the true data as a random sample drawn from a population
$\Omega $
according to some unknown density. We then replace
$\Omega $
by a much smaller random subset
$\Omega ^{\ast}$
, which we sample according to some known density. We generate synthetic data on the reduced space
$\Omega ^{\ast}$
by fitting the specified linear statistics obtained from the true data. To ensure privacy we use the common Laplacian mechanism. Employing the concept of Rényi condition number, which measures how well the sampling distribution is correlated with the population distribution, we derive explicit bounds on the privacy and accuracy provided by the proposed method.