PLAN: Variance-Aware Private Mean Estimation.

Differentially private mean estimation is an important building block in privacy-preserving algorithms for data analysis and machine learning. Though the trade-off between privacy and utility is well understood in the worst case, many datasets exhibit structure that could potentially be exploited to yield better algorithms. In this paper we present Private Limit Adapted Noise (plan), a family of differentially private algorithms for mean estimation in the setting where inputs are independently sampled from a distribution D over R𝑑, with coordinate-wise standard deviations 𝝈 ∈ R𝑑. Similar to mean estimation under Mahalanobis distance, plan tailors the shape of the noise to the shape of the data, but unlike previous algorithms the privacy budget is spent non-uniformly over the coordinates. Under a concentration assumption on D, we show how to exploit skew in the vector 𝝈, obtaining a (zero-concentrated) differentially private mean estimate with ℓ2 error proportional to ∥𝝈 ∥1. Previous work has either not taken 𝝈 into account, or measured error in Mahalanobis distance — in both cases resulting in ℓ2 error proportional to √𝑑 ∥𝝈 ∥2, which can be up to a factor √𝑑 larger. To verify the effectiveness of plan, we empirically evaluate accuracy on both synthetic and real-world data.