Making the Most of Parallel Composition in Differential Privacy
This work addresses the challenge of minimizing noise in differentially private query answering, which is crucial for data analysts and privacy practitioners, though it is incremental as it builds on existing graph-theoretic approximations.
The paper tackles the problem of optimizing parallel composition in differential privacy by introducing the concept of maximum overlap for queries, showing that it remains NP-hard for a class of predicate queries but can be approximated efficiently using graph-theoretic methods. The approach, applied to synthetic and real-world datasets, scales to large domain sizes and reduces noise in query answers by up to 60%.
We show that the `optimal' use of the parallel composition theorem corresponds to finding the size of the largest subset of queries that `overlap' on the data domain, a quantity we call the \emph{maximum overlap} of the queries. It has previously been shown that a certain instance of this problem, formulated in terms of determining the sensitivity of the queries, is NP-hard, but also that it is possible to use graph-theoretic algorithms, such as finding the maximum clique, to approximate query sensitivity. In this paper, we consider a significant generalization of the aforementioned instance which encompasses both a wider range of differentially private mechanisms and a broader class of queries. We show that for a particular class of predicate queries, determining if they are disjoint can be done in time polynomial in the number of attributes. For this class, we show that the maximum overlap problem remains NP-hard as a function of the number of queries. However, we show that efficient approximate solutions exist by relating maximum overlap to the clique and chromatic numbers of a certain graph determined by the queries. The link to chromatic number allows us to use more efficient approximate algorithms, which cannot be done for the clique number as it may underestimate the privacy budget. Our approach is defined in the general setting of $f$-differential privacy, which subsumes standard pure differential privacy and Gaussian differential privacy. We prove the parallel composition theorem for $f$-differential privacy. We evaluate our approach on synthetic and real-world data sets of queries. We show that the approach can scale to large domain sizes (up to $10^{20000}$), and that its application can reduce the noise added to query answers by up to 60\%.