# Gaussian Noise is Nearly Instance Optimal for Private Unbiased Mean Estimation

We investigate unbiased high-dimensional mean estimators in differential privacy. We consider differentially private mechanisms whose expected output equals the mean of the input dataset, for every dataset drawn from a fixed convex domain K in ℝ^d. In the setting of concentrated differential privacy, we show that, for every input such an unbiased mean estimator introduces approximately at least as much error as a mechanism that adds Gaussian noise with a carefully chosen covariance. This is true when the error is measured with respect to ℓ_p error for any p ≥ 2. We extend this result to local differential privacy, and to approximate differential privacy, but for the latter the error lower bound holds either for a dataset or for a neighboring dataset. We also extend our results to mechanisms that take i.i.d. samples from a distribution over K and are unbiased with respect to the mean of the distribution.

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