  We introduce a universal framework for characterizing the statistical
efficiency of a statistical estimation problem with differential privacy
guarantees. Our framework, which we call High-dimensional Propose-Test-Release
(HPTR), builds upon three crucial components: the exponential mechanism, robust
statistics, and the Propose-Test-Release mechanism. Gluing all these together
is the concept of resilience, which is central to robust statistical
estimation. Resilience guides the design of the algorithm, the sensitivity
analysis, and the success probability analysis of the test step in
Propose-Test-Release. The key insight is that if we design an exponential
mechanism that accesses the data only via one-dimensional robust statistics,
then the resulting local sensitivity can be dramatically reduced. Using
resilience, we can provide tight local sensitivity bounds. These tight bounds
readily translate into near-optimal utility guarantees in several cases. We
give a general recipe for applying HPTR to a given instance of a statistical
estimation problem and demonstrate it on canonical problems of mean estimation,
linear regression, covariance estimation, and principal component analysis. We
introduce a general utility analysis technique that proves that HPTR nearly
achieves the optimal sample complexity under several scenarios studied in the
literature.
