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Finding anonymization mechanisms to protect personal data is at the heart ofrecent machine learning research. Here, we consider the consequences of localdifferential privacy constraints on goodness-of-fit testing, i.e. thestatistical problem assessing whether sample points are generated from a fixeddensity $f_0$, or not. The observations are kept hidden and replaced by astochastic transformation satisfying the local differential privacy constraint.In this setting, we propose a testing procedure which is based on an estimationof the quadratic distance between the density $f$ of the unobserved samples and$f_0$. We establish an upper bound on the separation distance associated withthis test, and a matching lower bound on the minimax separation rates oftesting under non-interactive privacy in the case that $f_0$ is uniform, indiscrete and continuous settings. To the best of our knowledge, we provide thefirst minimax optimal test and associated private transformation under a localdifferential privacy constraint over Besov balls in the continuous setting,quantifying the price to pay for data privacy. We also present a test that isadaptive to the smoothness parameter of the unknown density and remains minimaxoptimal up to a logarithmic factor. Finally, we note that our results can betranslated to the discrete case, where the treatment of probability vectors isshown 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 thediscrete cases.

Minimax optimal goodness-of-fit testing for densities and multinomials under a local differential privacy constraint