Extremal Mechanisms for Local Differential Privacy (1407.1338v3)

Abstract: Local differential privacy has recently surfaced as a strong measure of privacy in contexts where personal information remains private even from data analysts. Working in a setting where both the data providers and data analysts want to maximize the utility of statistical analyses performed on the released data, we study the fundamental trade-off between local differential privacy and utility. This trade-off is formulated as a constrained optimization problem: maximize utility subject to local differential privacy constraints. We introduce a combinatorial family of extremal privatization mechanisms, which we call staircase mechanisms, and show that it contains the optimal privatization mechanisms for a broad class of information theoretic utilities such as mutual information and $f$-divergences. We further prove that for any utility function and any privacy level, solving the privacy-utility maximization problem is equivalent to solving a finite-dimensional linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has a number of variables that is exponential in the size of the alphabet the data lives in. To account for this, we show that two simple privatization mechanisms, the binary and randomized response mechanisms, are universally optimal in the low and high privacy regimes, and well approximate the intermediate regime.

• The paper introduces staircase mechanisms that achieve optimal utility across various information-theoretic metrics under local differential privacy constraints.
• It reduces the utility-privacy trade-off to a finite-dimensional linear program, establishing a rigorous framework for optimal privatization.
• Simplified binary and randomized response mechanisms are identified for high and low privacy settings, offering practical solutions for efficient data analysis.

Overview of "Extremal Mechanisms for Local Differential Privacy"

Introduction

This paper explores optimizing the balance between privacy and utility in the context of local differential privacy (LDP). LDP is a privacy framework where data is anonymized at the source, ensuring that even the data collector does not have access to the raw data. The paper focuses on maximizing the utility of statistical analyses while adhering to LDP constraints, casting this as a constrained optimization problem.

Main Contributions

Staircase Mechanisms

A key contribution is the identification of a family of mechanisms, named "staircase mechanisms", which provide optimal solutions for a variety of utility functions. The mechanisms are shown to achieve optimal privatization for numerous information-theoretic utilities, such as mutual information and $f$-divergences. Notably, the paper demonstrates that any utility-privacy trade-off can be reduced to solving a finite-dimensional linear program, where staircase mechanisms emerge as optimal solutions.

Computational Challenges and Simplified Mechanisms

Though solving the linear program can be computationally intensive due to its exponential nature in the data alphabet size, the authors propose that two mechanisms—the binary mechanism and the randomized response mechanism—achieve optimal utility in low and high privacy regimes, respectively, and offer a good approximation in intermediate regimes.

Theoretical and Practical Implications

The theoretical insights provided by these extremal mechanisms pave the way for constructing effective privacy-preserving algorithms. Practically, the findings suggest that in high privacy settings ($\varepsilon \leq \varepsilon^*$), utilizing a simple binary mechanism suffices. In low privacy settings, randomized responses become optimal. These results facilitate the implementation of privacy-preserving methods without sacrificing utility.

Extensions and Speculative Ideas

The work encapsulates the essential trade-offs and optimizations but also opens avenues for further research. Future exploration could extend beyond binary and randomized mechanisms. Moreover, investigating more complex interactions between individual data elements and their correlation could refine these mechanisms further.

Conclusion

The paper offers a rigorous yet applicable framework for balancing local differential privacy with the utility of data. By establishing that staircase mechanisms contain the optimal solutions and presenting simplified yet efficient approaches, this research significantly advances practical implementations of privacy-preserving data analysis.