  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.
