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Local differential privacy has recently received increasing attention fromthe statistics community as a valuable tool to protect the privacy ofindividual data owners without the need of a trusted third party. Similar tothe classical notion of randomized response, the idea is that data ownersrandomize their true information locally and only release the perturbed data.Many different protocols for such local perturbation procedures can bedesigned. In most estimation problems studied in the literature so far,however, no significant difference in terms of minimax risk between purelynon-interactive protocols and protocols that allow for some amount ofinteraction between individual data providers could be observed. In this paperwe show that for estimating the integrated square of a density, sequentiallyinteractive procedures improve substantially over the best possiblenon-interactive procedure in terms of minimax rate of estimation. Inparticular, in the non-interactive scenario we identify an elbow in the minimaxrate at $s=\frac34$, whereas in the sequentially interactive scenario the elbowis at $s=\frac12$. This is markedly different from both, the case of directobservations, where the elbow is well known to be at $s=\frac14$, as well asfrom the case where Laplace noise is added to the original data, where an elbowat $s= \frac94$ is obtained. We also provide adaptive estimators that achievethe optimal rate up to log-factors, we draw connections to non-parametricgoodness-of-fit testing and estimation of more general integral functionals andconduct a series of numerical experiments. The fact that a particular locallydifferentially private, but interactive, mechanism improves over the simplenon-interactive one is also of great importance for practical implementationsof local differential privacy.

Interactive versus non-interactive locally differentially private estimation: Two elbows for the quadratic functional