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