Efficient Computation of the Privacy Funnel: An Expectation-Maximization Relaxed Approach

Core Concepts

A novel relaxation variant of the Privacy Funnel (PF) problem is proposed, which is proven to be equivalent to the original problem in terms of optimal solutions and values. An accurate Alternating Expectation Minimization (AEM) algorithm is developed, with theoretically guaranteed convergence.

Abstract

The paper proposes a novel approach to solve the Privacy Funnel (PF) problem, which aims to release useful data while limiting the exposure of associated sensitive information. The key contributions are: A relaxation variant of the PF problem is derived by replacing the original objective function with an upper bound under the Expectation-Maximization (EM) framework. This relaxed model is proven to be equivalent to the original PF problem in terms of optimal solutions and values. Based on the relaxed model, an Alternating Expectation Minimization (AEM) algorithm is developed, which updates the primal variables and dual variables in an alternating manner. The closed-form expressions for the updates ensure high computational efficiency. The descent of the objective function is theoretically estimated, and the convergence of the iterative sequence to a Karush-Kuhn-Tucker (KKT) point is guaranteed through Pinsker's inequality. Numerical experiments on synthetic and real-world datasets demonstrate the effectiveness of the proposed AEM algorithm, outperforming existing methods in terms of accuracy and stability.

Stats

The paper reports the following key figures: The synthetic conditional distribution PS|X used in the experiments. The uniform and non-uniform distributions PX used in the experiments. The cardinality of the variables S, X, and Y in the real-world datasets.

Quotes

"The privacy funnel (PF) gives a framework of privacy-preserving data release, where the goal is to release useful data while also limiting the exposure of associated sensitive information." "Research related to the PF model has covered a variety of setups in information theory, machine learning and other fields." "The closed-form solutions ensure the efficiency of our algorithm. Moreover, the descent of objective is theoretically estimated, and the convergence of iterative sequence to a Karush-Kuhn-Tucker (KKT) point is guaranteed by the Pinsker's inequality."

Key Insights Distilled From

An Expectation-Maximization Relaxed Method for Privacy Funnel

by Lingyi Chen,... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00616.pdf

Deeper Inquiries

How can the proposed AEM algorithm be extended to handle continuous variables in the Privacy Funnel problem

To extend the Alternating Expectation Minimization (AEM) algorithm to handle continuous variables in the Privacy Funnel problem, we need to adapt the optimization process to accommodate the characteristics of continuous data. One approach is to introduce probability density functions for the continuous variables and modify the update rules accordingly. Representation of Continuous Variables: For continuous variables, we can represent the conditional distributions using probability density functions. Instead of discrete probabilities, we would work with continuous distributions to model the relationships between the variables. Update Rules: The update rules for continuous variables would involve calculus operations to optimize the objective function. For example, the update of the primal variables would require derivatives with respect to the continuous variables and solving equations involving integrals. Convergence Analysis: The convergence analysis of the algorithm would need to consider the properties of continuous functions, such as differentiability and integrability. The descent estimation would involve gradients and integrals instead of discrete summations. Numerical Stability: Ensuring numerical stability in handling continuous variables is crucial. Techniques like regularization and adaptive step sizes may be employed to prevent numerical issues during optimization. By adapting the AEM algorithm to handle continuous variables, we can effectively address Privacy Funnel problems involving continuous data distributions, expanding the applicability of the algorithm to a wider range of scenarios.

What are the potential applications of the Privacy Funnel framework beyond data release, such as in machine learning or other domains

The Privacy Funnel framework, beyond its primary application in privacy-preserving data release, has various potential applications in machine learning and other domains. Some of these applications include: Machine Learning: The Privacy Funnel framework can be utilized in machine learning tasks where balancing privacy and utility is crucial. For instance, in federated learning settings, where data privacy is a concern, the Privacy Funnel can help in optimizing models while preserving the privacy of individual data contributors. Anomaly Detection: In anomaly detection systems, the Privacy Funnel can be used to filter out sensitive information while retaining relevant features for anomaly detection. This ensures that the detection process does not compromise the privacy of individuals. Healthcare: In healthcare analytics, the Privacy Funnel can aid in sharing medical data for research purposes while protecting patient privacy. It allows for the extraction of useful insights from medical records without revealing sensitive patient information. Financial Data Analysis: In financial data analysis, the Privacy Funnel can be applied to share financial information securely among institutions for collaborative analysis. It ensures that confidential financial details are not exposed during data sharing processes. Recommendation Systems: Privacy Funnel techniques can enhance recommendation systems by enabling personalized recommendations without compromising user privacy. It can help in filtering out sensitive user data while improving the accuracy of recommendations. By exploring these diverse applications, the Privacy Funnel framework can contribute significantly to various fields beyond traditional data release scenarios.

Can the relaxation technique used in this work be applied to solve other non-convex optimization problems in information theory or machine learning

The relaxation technique used in this work, based on Jensen's inequality and the derivation of upper bounds, can indeed be applied to solve other non-convex optimization problems in information theory and machine learning. Here are some key points on how this relaxation technique can be beneficial in solving a broader range of optimization problems: Non-Convex Optimization: Many optimization problems in machine learning and information theory are inherently non-convex, posing challenges for finding global optima. By introducing relaxation techniques similar to the one proposed in this work, it becomes possible to derive tractable optimization formulations that are easier to solve. Convergence Guarantees: The relaxation technique provides a way to ensure convergence and stability in iterative optimization algorithms. By establishing equivalence between the original problem and its relaxed variant, one can develop algorithms with theoretical convergence guarantees, as demonstrated in this work. Efficient Algorithms: The relaxation approach allows for the development of efficient algorithms that involve closed-form iterations, reducing computational complexity and improving convergence speed. This can be particularly useful in large-scale optimization problems where computational efficiency is crucial. Generalizability: The principles of relaxation and upper bound derivation can be extended to various optimization problems beyond the Privacy Funnel framework. By adapting these techniques to different problem settings, researchers can address a wide range of non-convex optimization challenges in diverse domains. Overall, the relaxation technique employed in this work opens up possibilities for tackling complex non-convex optimization problems in information theory and machine learning, offering a structured approach to optimization with theoretical guarantees and computational efficiency.