We consider the problem of nonparametric density estimation under privacy constraints in an adversarial framework. To this end, we study minimax rates under local differential privacy over Sobolev spaces. We first obtain a lower bound which allows us to quantify the impact of privacy compared with the classical framework. Next, we introduce a new Coordinate block privacy mechanism that guarantees local differential privacy, which, coupled with a projection estimator, achieves the minimax optimal rates.

翻译：我们在对抗框架下研究隐私约束下的非参数密度估计问题。为此，我们分析了Sobolev空间上局部差分隐私约束下的极小极大速率。首先推导出一个下界，该下界使我们能够量化隐私保护相较于经典框架的影响。随后，我们提出一种新型坐标块隐私机制以保证局部差分隐私，该机制与投影估计量相结合，实现了极小极大最优速率。