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.

本文研究了在Lipschitz和Sobolev空间中具有全局隐私性的非参数密度估计问题，并探讨了隐私预算不应为常数的情况下的最优解。 结果表明投影估计器在这种新的设置下与纯差分隐私的相同密度类别几乎是最优的。 

全球隐私成本和密度估计