Unbounded Differentially Private Quantile and Maximum Estimation

In this work we consider the problem of differentially private computation of quantiles for the data, especially the highest quantiles such as maximum, but with an unbounded range for the dataset. We show that this can be done efficiently through a simple invocation of $\texttt{AboveThreshold}$, a subroutine that is iteratively called in the fundamental Sparse Vector Technique, even when there is no upper bound on the data. In particular, we show that this procedure can give more accurate and robust estimates on the highest quantiles with applications towards clipping that is essential for differentially private sum and mean estimation. In addition, we show how two invocations can handle the fully unbounded data setting. Within our study, we show that an improved analysis of $\texttt{AboveThreshold}$ can improve the privacy guarantees for the widely used Sparse Vector Technique that is of independent interest. We give a more general characterization of privacy loss for $\texttt{AboveThreshold}$ which we immediately apply to our method for improved privacy guarantees. Our algorithm only requires one $O(n)$ pass through the data, which can be unsorted, and each subsequent query takes $O(1)$ time. We empirically compare our unbounded algorithm with the state-of-the-art algorithms in the bounded setting. For inner quantiles, we find that our method often performs better on non-synthetic datasets. For the maximal quantiles, which we apply to differentially private sum computation, we find that our method performs significantly better.