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

Finding anonymization mechanisms to protect personal data is at the heart of 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 hidden and replaced by a stochastic transformation satisfying the local differential privacy constraint. In this setting, we propose a new testing procedure which is based on an estimation of the quadratic distance between the density $f$ of the unobserved sample and $f_0$. We establish minimax separation rates for our test over Besov balls. We also provide a lower bound, proving the optimality of our result. To the best of our knowledge, we provide the first minimax optimal test and associated private transformation under a local differential privacy constraint, quantifying the price to pay for data privacy.