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New Lower Bounds for Private Estimation and\\a Generalized Fingerprinting Lemma. (arXiv:2205.08532v1 [cs.DS])
May 18, 2022, 1:20 a.m. | Gautam Kamath, Argyris Mouzakis, Vikrant Singhal
cs.CR updates on arXiv.org arxiv.org
We prove new lower bounds for statistical estimation tasks under the
constraint of $\paren{\eps, \delta}$-differential privacy. First, we provide
tight lower bounds for private covariance estimation of Gaussian distributions.
We show that estimating the covariance matrix in Frobenius norm requires
$\Omega\paren{d^2}$ samples, and in spectral norm requires
$\Omega\paren{d^{\frac{3}{2}}}$ samples, both matching upper bounds up to
logarithmic factors. We prove these bounds via our main technical contribution,
a broad generalization of the fingerprinting method~\cite{BunUV14} to
exponential families. Additionally, using the private …
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