Gradual Release of Sensitive Data under Differential Privacy

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We introduce the problem of releasing private data under differential privacy when the privacy level is subject to change over time. Existing work assumes that privacy level is determined by the system designer as a fixed value before private data is released. For certain applications, however, users may wish to relax the privacy level for subsequent releases of the same data after either a re-evaluation of the privacy concerns or the need for better accuracy. Specifically, given a database containing private data, we assume that a response y1 that preserves \( \epsilon _1\)-differential privacy has already been published. Then, the privacy level is relaxed to \( \epsilon _2\), with \( \epsilon _2 > \epsilon _1\), and we wish to publish a more accurate response y2 while the joint response (y1,y2) preserves \( \epsilon _2\)-differential privacy. How much accuracy is lost in the scenario of gradually releasing two responses y1 and y2 compared to the scenario of releasing a single response that is \( \epsilon _2\)-differentially private? Our
 results consider the more general case with multiple privacy level relaxations and show that there exists a composite mechanism that achieves no loss
 in accuracy.
 We consider the case in which the private data lies within Rn with an adjacency relation induced by the \( \ell _1\)-norm, and we initially focus on mechanisms that approximate identity queries. We show that the same accuracy can be achieved in the case of gradual release through a mechanism whose outputs can be described by a lazy Markov stochastic process. This stochastic process has a closed form expression and can be efficiently sampled. Moreover, our results extend beyond identity queries to a more general family of privacy-preserving mechanisms. To this end, we demonstrate the applicability of our tool to multiple scenarios including Google’s project RAPPOR, trading of private data, and controlled transmission of private data in a social network. Finally, we derive similar results for the approximated differential privacy.