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：我们研究在Lipschitz和Sobolev空间下，以及集中隐私约束下的非参数密度估计问题。特别地，我们探究隐私预算不设为常数的情形。我们考虑经典的集中差分隐私定义，同时也关注较新的集中集中差分隐私概念。我们复现了Barber和Duchi（2014）的结论，即在常规差分隐私下，对于L2风险而言，直方图估计器在Lipschitz分布上是最优的，并将其推广到其他范数和隐私概念上。随后，我们考察更高阶的光滑性，得出两点结论：第一，与恒定隐私预算下的情况（Wasserman和Zhou，2010）相反，存在一些情形，施加隐私保护会降低Sobolev密度估计的常规极小化极大风险；第二，在纯差分隐私的新设置下，所谓的投影估计器针对同样类别的密度是接近最优的，但与恒定隐私预算情形不同，这需要以放松条件为代价。在零集中差分隐私下，则无需放松条件，我们证明此时估计是最优的。