  The challenge of producing accurate statistics while respecting the privacy
of the individuals in a sample is an important area of research. We study
minimax lower bounds for classes of differentially private estimators. In
particular, we show how to characterize the power of a statistical test under
differential privacy in a plug-and-play fashion by solving an appropriate
transport problem. With specific coupling constructions, this observation
allows us to derive Le Cam-type and Fano-type inequalities not only for regular
definitions of differential privacy but also for those based on Renyi
divergence. We then proceed to illustrate our results on three simple, fully
worked out examples. In particular, we show that the problem class has a huge
importance on the provable degradation of utility due to privacy. In certain
scenarios, we show that maintaining privacy results in a noticeable reduction
in performance only when the level of privacy protection is very high.
Conversely, for other problems, even a modest level of privacy protection can
lead to a significant decrease in performance. Finally, we demonstrate that the
DP-SGLD algorithm, a private convex solver, can be employed for maximum
likelihood estimation with a high degree of confidence, as it provides
near-optimal results with respect to both the size of the sample and the level
of privacy protection. This algorithm is applicable to a broad range of
parametric estimation procedures, including exponential families.
