We study non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under global privacy. In particular, we investigate regimes where the privacy budget is not supposed to be constant. We consider the classical definition of global differential privacy, but also the more recent notion of global 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：我们研究了在全局隐私保护下，针对Lipschitz和Sobolev空间中密度函数的非参数密度估计。特别地，我们考察了隐私预算不为常数的情形。我们不仅考虑了经典的全局差分隐私定义，还引入了较新的全局集中差分隐私概念。我们复现了Barber与Duchi（2014）的结论，该结论指出直方图估计器在L2风险下对Lipschitz分布族是最优的，且满足标准差分隐私；我们将其推广到其他范数和隐私定义。接着，我们探究了更高光滑度的情况，得出两个结论：第一，与恒定隐私预算情形（Wasserman与Zhou，2010）相反，存在某些情形下施加隐私保护会降低Sobolev密度估计的常规极小化风险；第二，在此新设定下，带有纯差分隐私的所谓投影估计器对相同密度族接近最优，但与恒定隐私预算情况不同，这需要以放松条件为代价。在使用零集中差分隐私时，则无需放松条件，且我们证明此时的估计是最优的。