Minimax rate for multivariate data under componentwise local differential privacy constraints

Our research delves into the balance between maintaining privacy and
preserving statistical accuracy when dealing with multivariate data that is
subject to \textit{componentwise local differential privacy} (CLDP). With CLDP,
each component of the private data is made public through a separate privacy
channel. This allows for varying levels of privacy protection for different
components or for the privatization of each component by different entities,
each with their own distinct privacy policies. We develop general techniques
for establishing minimax bounds that shed light on the statistical cost of
privacy in this context, as a function of the privacy levels $\alpha_1, ... ,
\alpha_d$ of the $d$ components. We demonstrate the versatility and efficiency
of these techniques by presenting various statistical applications.
Specifically, we examine nonparametric density and covariance estimation under
CLDP, providing upper and lower bounds that match up to constant factors, as
well as an associated data-driven adaptive procedure. Furthermore, we quantify
the probability of extracting sensitive information from one component by
exploiting the fact that, on another component which may be correlated with the
first, a smaller degree of privacy protection is guaranteed.