Universal Private Estimators
This work addresses the challenge of private statistical estimation for arbitrary unknown distributions, removing key limitations in differential privacy, though it is incremental in extending existing methods to broader settings.
The paper tackles the problem of estimating statistical moments like mean, variance, and scale under pure differential privacy without requiring prior boundedness assumptions on the distribution, achieving strong utility guarantees for arbitrary continuous distributions and matching or improving existing estimators for specific families like Gaussians.
We present \textit{universal} estimators for the statistical mean, variance, and scale (in particular, the interquartile range) under pure differential privacy. These estimators are universal in the sense that they work on an arbitrary, unknown continuous distribution $\mathcal{P}$ over $\mathbb{R}$, while yielding strong utility guarantees except for ill-behaved $\mathcal{P}$. For certain distribution families like Gaussians or heavy-tailed distributions, we show that our universal estimators match or improve existing estimators, which are often specifically designed for the given family and under \textit{a priori} boundedness assumptions on the mean and variance of $\mathcal{P}$. This is the first time these boundedness assumptions are removed under pure differential privacy. The main technical tools in our development are instance-optimal empirical estimators for the mean and quantiles over the unbounded integer domain, which can be of independent interest.