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Tight Bounds on the Contraction of Locally Differentially Private Mechanisms


Core Concepts
The authors derive tight upper bounds on the divergence between the output distributions of an ε-LDP mechanism under different f-divergences, including KL-divergence and χ2-divergence. These bounds are used to establish locally private versions of powerful information-theoretic tools for bounding minimax estimation risks.
Abstract
The paper investigates the contraction properties of locally differentially private (LDP) mechanisms. The authors derive tight upper bounds on the divergence between the output distributions of an ε-LDP mechanism under different f-divergences, including KL-divergence and χ2-divergence. The key technical results are: Theorem 1 shows that the contraction coefficients under KL-divergence, χ2-divergence, and squared Hellinger distance are upper bounded by (eε - 1)2 / (eε + 1)2, which is tight. Theorem 2 provides an upper bound on χ2(PK||QK) in terms of TV(P,Q) and ε, which is significantly tighter than previous bounds, especially for ε ≥ 1. The authors then leverage these technical results to develop a systematic framework for quantifying the cost of local privacy in several statistical problems, including: Locally private Fisher information and van Trees inequality (Lemma 1, Corollary 1) Improved private versions of Le Cam's and Assouad's methods (Theorems 3 and 4) Application to entropy estimation, distribution estimation, and non-parametric density estimation (Corollaries 2-4) Locally private mutual information method (Theorem 5) Locally private hypothesis testing (Lemma 2) The results demonstrate that the authors' technical contributions lead to tighter privacy analyses compared to the state-of-the-art in several statistical problems.
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Key Insights Distilled From

by Shahab Asood... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2210.13386.pdf
Contraction of Locally Differentially Private Mechanisms

Deeper Inquiries

How can the techniques developed in this paper be extended to other statistical problems beyond the ones considered

The techniques developed in this paper can be extended to other statistical problems beyond the ones considered by applying them to various estimation and inference tasks. For example, the framework of contraction bounds for locally differentially private mechanisms can be utilized in problems such as regression analysis, classification tasks, clustering algorithms, and anomaly detection. By incorporating the tight upper bounds on divergence between output distributions of LDP mechanisms, researchers can enhance the privacy guarantees and optimize the performance of these algorithms in a privacy-preserving manner. Additionally, the systemic framework developed in this paper can be adapted to address a wide range of statistical challenges, providing sharper privacy analyses and improved minimax estimation risks in diverse applications.

Are there any limitations or assumptions in the current framework that could be relaxed or generalized

While the current framework provides valuable insights into the contraction properties of locally differentially private mechanisms, there are certain limitations and assumptions that could be relaxed or generalized to broaden the applicability of the results. One potential area for improvement is relaxing the assumption of sequential interaction among users in the LDP mechanisms. By considering more flexible interaction patterns or non-sequential setups, the framework can be extended to accommodate a wider range of real-world scenarios where users may interact in different ways. Additionally, relaxing the assumption of a specific loss function or norm in the estimation tasks could make the results more versatile and applicable to a broader set of statistical problems. By generalizing the framework to handle different types of loss functions and norms, researchers can address a more diverse set of estimation challenges under local privacy constraints.

What are the potential applications of the tight contraction bounds for LDP mechanisms in areas beyond statistical estimation and hypothesis testing

The tight contraction bounds for locally differentially private mechanisms have the potential for various applications beyond statistical estimation and hypothesis testing. One key application area is in machine learning algorithms, where privacy-preserving techniques are crucial for protecting sensitive data. By incorporating these contraction bounds into machine learning models, researchers can develop more robust and secure algorithms that maintain privacy guarantees while achieving high performance. Additionally, the bounds can be applied in data analytics, privacy-preserving data sharing, secure multi-party computation, and other areas where privacy and accuracy are paramount. The results can also be utilized in privacy-enhancing technologies, secure data processing, and confidential computing to ensure data privacy and security in various domains.
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