We study non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under central privacy. In particular, we investigate regimes where the privacy budget is not supposed to be constant. We consider the classical definition of central differential privacy, but also the more recent notion of central concentrated differential privacy. We recover the result of Barber \& Duchi (2014) stating that histogram estimators are optimal against Lipschitz distributions for the L2 risk, and under regular differential privacy, and we extend it to other norms and notions of privacy. Then, we investigate higher degrees of smoothness, drawing two conclusions: First, and contrary to what happens with constant privacy budget (Wasserman \& Zhou, 2010), there are regimes where imposing privacy degrades the regular minimax risk of estimation on Sobolev densities. Second, so-called projection estimators are near-optimal against the same classes of densities in this new setup with pure differential privacy, but contrary to the constant privacy budget case, it comes at the cost of relaxation. With zero concentrated differential privacy, there is no need for relaxation, and we prove that the estimation is optimal.

翻译：我们研究了在Lipschitz和Sobolev空间以及中心隐私条件下的非参数密度估计。特别地，我们考察了隐私预算并非恒定不变的场景。我们考虑了中心差分隐私的经典定义，也涉及了更近期提出的中心集中差分隐私概念。我们重现了Barber & Duchi（2014）的结果，该结果指出在L2风险下，直方图估计器针对Lipschitz分布是最优的，且符合标准差分隐私条件，并将其推广到其他范数和隐私概念。随后，我们探究了更高阶的光滑性，得出两个结论：首先，与恒定隐私预算情况（Wasserman & Zhou, 2010）相反，在某些场景下引入隐私保护会降低针对Sobolev密度的常规极小极大估计风险。其次，所谓的投影估计器在纯差分隐私的新设定下针对同类密度族接近最优，但与恒定隐私预算情况不同，这需要以松弛为代价。在零集中差分隐私条件下，无需松弛，我们证明了该估计是最优的。