Better and Simpler Lower Bounds for Differentially Private Statistical Estimation. (arXiv:2310.06289v2 [math.ST] UPDATED)

We provide optimal lower bounds for two well-known parameter estimation (also
known as statistical estimation) tasks in high dimensions with approximate
differential privacy. First, we prove that for any $\alpha \le O(1)$,
estimating the covariance of a Gaussian up to spectral error $\alpha$ requires
$\tilde{\Omega}\left(\frac{d^{3/2}}{\alpha \varepsilon} +
\frac{d}{\alpha^2}\right)$ samples, which is tight up to logarithmic factors.
This result improves over previous work which established this for $\alpha \le
O\left(\frac{1}{\sqrt{d}}\right)$, and is also simpler than previous work.
Next, we prove that …

differential privacy error high math omega parameter privacy private prove spectral well-known