Extremal Mechanisms for Local Differential Privacy

Local differential privacy has recently surfaced as a strong measure of privacy in contexts where personal information remains private even from data analysts. Working in a setting where both the data providers and data analysts want to maximize the utility of statistical analyses performed on the released data, we study the fundamental trade-off between local differential privacy and utility. This trade-off is formulated as a constrained optimization problem: maximize utility subject to local differential privacy constraints. We introduce a combinatorial family of extremal privatization mechanisms, which we call staircase mechanisms, and show that it contains the optimal privatization mechanisms for a broad class of information theoretic utilities such as mutual information and $f$-divergences. We further prove that for any utility function and any privacy level, solving the privacy-utility maximization problem is equivalent to solving a finite-dimensional linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has a number of variables that is exponential in the size of the alphabet the data lives in. To account for this, we show that two simple privatization mechanisms, the binary and randomized response mechanisms, are universally optimal in the low and high privacy regimes, and well approximate the intermediate regime.Comment: 52 pages, 10 figures in JMLR 201

p-probabilistic k-anonymous microaggregation for the anonymization of surveys with uncertain participation

We develop a probabilistic variant of k-anonymous microaggregation which we term p-probabilistic resorting to a statistical model of respondent participation in order to aggregate quasi-identifiers in such a manner that k-anonymity is concordantly enforced with a parametric probabilistic guarantee. Succinctly owing the possibility that some respondents may not finally participate, sufficiently larger cells are created striving to satisfy k-anonymity with probability at least p. The microaggregation function is designed before the respondents submit their confidential data. More precisely, a specification of the function is sent to them which they may verify and apply to their quasi-identifying demographic variables prior to submitting the microaggregated data along with the confidential attributes to an authorized repository. We propose a number of metrics to assess the performance of our probabilistic approach in terms of anonymity and distortion which we proceed to investigate theoretically in depth and empirically with synthetic and standardized data. We stress that in addition to constituting a functional extension of traditional microaggregation, thereby broadening its applicability to the anonymization of statistical databases in a wide variety of contexts, the relaxation of trust assumptions is arguably expected to have a considerable impact on user acceptance and ultimately on data utility through mere availability.Peer ReviewedPostprint (author's final draft