We provide the first $\widetilde{\mathcal{O}}(d)$-sample algorithm for sampling from unbounded Gaussian distributions under the constraint of $(\varepsilon, \delta)$-differential privacy. This is a quadratic improvement over previous results for the same problem, settling an open question of Ghazi, Hu, Kumar, and Manurangsi.

We study person-level differentially private (DP) mean estimation in the case where each person holds multiple samples. DP here requires the usual notion of distributional stability when all of a person’s datapoints can be modified. Informally, if n people each have m samples from an unknown d-dimensional distribution with bounded k-th moments, we show thatpeople are necessary and sufficient to estimate the mean up to distance α in ℓ2-norm under ε-differential privacy (and its common relaxations). In the multivariate setting, we give computationally efficient algorithms under approximate DP and computationally inefficient algorithms under pure DP, and our nearly matching lower bounds hold for the most permissive case of approximate DP. Our computationally efficient estimators are based on the standard clip-and-noise framework, but the analysis for our setting requires both new algorithmic techniques and new analyses. In particular, our new bounds on the tails of sums of independent, vector-valued, bounded-moments random variables may be of interest.

We give an example of a class of distributions that is learnable in total variation distance with a finite number of samples, but not learnable under $(\varepsilon, \delta)$-differential privacy. This refutes a conjecture of Ashtiani.

We give the first polynomial-time, polynomial-sample, differentially private estimator for the mean and covariance of an arbitrary Gaussian distribution $N(\mu,\Sigma)$ in $\R^d$. All previous estimators are either nonconstructive, with unbounded running time, or require the user to specify a priori bounds on the parameters $\mu$ and $\Sigma$. The primary new technical tool in our algorithm is a new differentially private preconditioner that takes samples from an arbitrary Gaussian $N(0,\Sigma)$ and returns a matrix $A$ such that $A \Sigma A^T$ has constant condition number

We prove new lower bounds for statistical estimation tasks under the constraint of $(\varepsilon,\delta)$-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 $\Omega(d^2)$ samples, and in spectral norm requires $\Omega(d^{3/2})$ 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 to exponential families. Additionally, using the private Assouad method of Acharya, Sun, and Zhang, we show a tight $\Omega(d/(\alpha^2 \varepsilon))$ lower bound for estimating the mean of a distribution with bounded covariance to $\alpha$-error in $\ell_2$-distance. Prior known lower bounds for all these problems were either polynomially weaker or held under the stricter condition of $(\varepsilon,0)$-differential privacy.