Iterative Constructions and Private Data Release (1107.3731v2)

Abstract: In this paper we study the problem of approximately releasing the cut function of a graph while preserving differential privacy, and give new algorithms (and new analyses of existing algorithms) in both the interactive and non-interactive settings. Our algorithms in the interactive setting are achieved by revisiting the problem of releasing differentially private, approximate answers to a large number of queries on a database. We show that several algorithms for this problem fall into the same basic framework, and are based on the existence of objects which we call iterative database construction algorithms. We give a new generic framework in which new (efficient) IDC algorithms give rise to new (efficient) interactive private query release mechanisms. Our modular analysis simplifies and tightens the analysis of previous algorithms, leading to improved bounds. We then give a new IDC algorithm (and therefore a new private, interactive query release mechanism) based on the Frieze/Kannan low-rank matrix decomposition. This new release mechanism gives an improvement on prior work in a range of parameters where the size of the database is comparable to the size of the data universe (such as releasing all cut queries on dense graphs). We also give a non-interactive algorithm for efficiently releasing private synthetic data for graph cuts with error O(|V|^{1.5}). Our algorithm is based on randomized response and a non-private implementation of the SDP-based, constant-factor approximation algorithm for cut-norm due to Alon and Naor. Finally, we give a reduction based on the IDC framework showing that an efficient, private algorithm for computing sufficiently accurate rank-1 matrix approximations would lead to an improved efficient algorithm for releasing private synthetic data for graph cuts. We leave finding such an algorithm as our main open problem.

• The paper presents a novel iterative database construction (IDC) framework that unifies and improves private query release mechanisms for graph cut functions.
• It introduces an IDC algorithm based on Frieze/Kannan low-rank matrix decomposition to optimize query responses in dense graph scenarios.
• The non-interactive method employs randomized response to deliver synthetic graph data with an error bound of O(|V|^1.5), enhancing prior efficiency benchmarks.

Overview of Iterative Constructions and Private Data Release

The paper by Gupta, Roth, and ULLMan addresses the challenge of differentially private data release, specifically focusing on the problem of approximately releasing the cut function of a graph. This exploration is carried out within both interactive and non-interactive settings. The work is situated at the intersection of privacy-preserving data analysis and algorithms for querying databases.

Interactive Setting

In the interactive setting, the authors revisit the issue of providing differentially private answers to a plethora of queries on a database. They unify several algorithms under a generic framework centered on iterative database construction (IDC) algorithms. The concept of IDC is presented as a modular framework that allows for the derivation of new private query release mechanisms. The framework simplifies and enhances the analysis of past algorithms, leading to improved bounds on query release mechanisms.

To exemplify the use of this framework, the paper introduces a new IDC algorithm based on the Frieze/Kannan low-rank matrix decomposition. This new mechanism optimizes previous work in contexts where the database size is on par with the size of the data universe, facilitating the release of all cut queries in dense graphs with improved efficiency.

Non-Interactive Setting

The authors also tackle the problem of releasing private synthetic data for graph cuts in a non-interactive manner. Their proposed algorithm uses randomized response and implements a non-private version of a constant-factor approximation algorithm for the cut-norm from the work of Alon and Naor. The result is an efficient algorithm achieving error $O(|V|^{1.5})$, significantly reducing prior bounds. Furthermore, the paper illustrates a reduction based on the IDC framework, suggesting that an efficient private rank-1 matrix approximation algorithm could yield superior efficiency for synthetic data release.

Implications and Future Directions

The implications of this work are both practical and theoretical. From a practical perspective, the introduction of a new framework for private query release mechanisms can be seamlessly adapted and extended to diverse applications involving large databases. Theoretically, the findings open up further exploration into low-rank decompositions and their role in achieving privacy guarantees.

A pivotal open problem remains: developing an efficient, private algorithm for good rank-1 approximations. Solving this problem could significantly enhance the efficiency of current private data release mechanisms, as highlighted by the potential impact on synthetic data for graph cuts. Future research may also explore whether similar frameworks can be generalized to cover more complex data structures and query types beyond graph cuts and linear queries.

Conclusively, this paper extends the landscape of differential privacy by providing robust mechanisms for private data analysis while also setting the stage for future explorations in efficient algorithm design under privacy constraints.