Local differential privacy has been proposed as a strong measure of privacy under data collection scenarios, where individuals are willing to share their data but are concerned about revealing sensitive information. Both data providers and data collectors want to maximize the benefit of statistical inference performed on the data but at the same time need to protect the privacy of the participating individuals. We address a general problem of utility maximization under local differential privacy. Our main result is a characterization of the combinatorial structure of the optimal solution resulting in a family of extremal mechanisms we call staircase mechanisms. These data dependent mechanisms add to the few basic privatization schemes known in the literature (eg: the exponential mechanism and noise-adding mechanisms). Finally, we show that two simple staircase mechanisms (the binary and randomized response mechanisms) are optimal in the high and low privacy regimes, respectively. Our results also characterize lower bounds in differential privacy, providing some new lower bounds as well as recovering some known results. As a motivating example, we provide optimal mechanisms and show that the effective sample size reduces from n to ε 2 n under differential privacy in the context of hypothesis testing.
For example, consider providing ε-local differential privacy on the source of a transaction REF .
Extremal Mechanisms for Local Differential Privacy
ArXiv
ArXiv
Computer Science Mathematics