  Finding anonymization mechanisms to protect personal data is at the heart of
recent machine learning research. Here, we consider the consequences of local
differential privacy constraints on goodness-of-fit testing, i.e. the
statistical problem assessing whether sample points are generated from a fixed
density $f_0$, or not. The observations are kept hidden and replaced by a
stochastic transformation satisfying the local differential privacy constraint.
In this setting, we propose a testing procedure which is based on an estimation
of the quadratic distance between the density $f$ of the unobserved samples and
$f_0$. We establish an upper bound on the separation distance associated with
this test, and a matching lower bound on the minimax separation rates of
testing under non-interactive privacy in the case that $f_0$ is uniform, in
discrete and continuous settings. To the best of our knowledge, we provide the
first minimax optimal test and associated private transformation under a local
differential privacy constraint over Besov balls in the continuous setting,
quantifying the price to pay for data privacy. We also present a test that is
adaptive to the smoothness parameter of the unknown density and remains minimax
optimal up to a logarithmic factor. Finally, we note that our results can be
translated to the discrete case, where the treatment of probability vectors is
shown to be equivalent to that of piecewise constant densities in our setting.
That is why we work with a unified setting for both the continuous and the
discrete cases.
