Differentially Private Histogram, Predecessor, and Set Cardinality under Continual Observation

Abstract

Differential privacy is the de-facto privacy standard in data analysis. The classic model of differential privacy considers the data to be static. The dynamic setting, called differential privacy under continual observation, captures many applications more realistically. In this work we consider several natural dynamic data structure problems under continual observation, where we want to maintain information about a changing data set such that we can answer certain sets of queries at any given time while satisfying $\epsilon$-differential privacy. The problems we consider include (a) maintaining a histogram and various extensions of histogram queries such as quantile queries, (b) maintaining a predecessor search data structure of a dynamically changing set in a given ordered universe, and (c) maintaining the cardinality of a dynamically changing set. For (a) we give new error bounds parameterized in the maximum output of any query $c_{\max}$: our algorithm gives an upper bound of $O(d\log^2dc_{\max}+\log T)$ for computing histogram, the maximum and minimum column sum, quantiles on the column sums, and related queries. The bound holds for unknown $c_{\max}$ and $T$. For (b), we give a general reduction to orthogonal range counting. Further, we give an improvement for the case where only insertions are allowed. We get a data structure which for a given query, returns an interval that contains the predecessor, and at most $O(\log^2 u \sqrt{\log T})$ more elements, where $u$ is the size of the universe. The bound holds for unknown $T$. Lastly, for (c), we give a parameterized upper bound of $O(\min(d,\sqrt{K\log T}))$, where $K$ is an upper bound on the number of updates. We show a matching lower bound. Finally, we show how to extend the bound for (c) for unknown $K$ and $T$.