Dimension-Free Private Mean Estimation is Impossible Without the Covariance

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Abstract

We study the limits of differentially private Gaussian mean estimation when the covariance matrix is unknown. Prior work shows that when the covariance $Σ$ is known, the optimal sample complexity can depend only on its spectrum, rather than on the ambient dimension $d$. We establish new lower bounds for Gaussian mean estimation with Euclidean error which show that private estimators cannot achieve similar guarantees when the covariance is unknown. First, when the sample size $n$ is smaller than the effective dimension $\mathrm{trace}(Σ)/\|Σ\|_{\text{op}}$, the optimal error when the covariance is unknown can be larger than the optimal error when it is known by a factor of $\sqrt{d/n}$. Second, when the sample size is larger than the effective dimension, even for diagonal covariance matrices with effective dimension bounded by $\log d$, the optimal error when the covariance is unknown can be larger than the optimal error when it is known by a factor of $\sqrt{d}/n^{3/2}$. Together, these results show that in the unknown covariance setting, exploiting anisotropic structure under differential privacy can require a dimension-dependent error or sample complexity.

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