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Register a new userImproved Matrix Gaussian Mechanism for Differential Privacy
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Jungang Yang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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The wide deployment of machine learning in recent years gives rise to a great demand for large-scale and high-dimensional data, for which the privacy raises serious concern. Differential privacy (DP) mechanisms are conventionally developed for scalar values, not for structural data like matrices. Our work proposes Improved Matrix Gaussian Mechanism (IMGM) for matrix-valued DP, based on the necessary and sufficient condition of $ (varepsilon,delta) $-differential privacy. IMGM only imposes constraints on the singular values of the covariance matrices of the noise, which leaves room for design. Among the legitimate noise distributions for matrix-valued DP, we find the optimal one turns out to be i.i.d. Gaussian noise, and the DP constraint becomes a noise lower bound on each element. We further derive a tight composition method for IMGM. Apart from the theoretical analysis, experiments on a variety of models and datasets also verify that IMGM yields much higher utility than the state-of-the-art mechanisms at the same privacy guarantee.
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Differential privacy mechanism design has traditionally been tailored for a scalar-valued query function. Although many mechanisms such as the Laplace and Gaussian mechanisms can be extended to a matrix-valued query function by adding i.i.d. noise to each element of the matrix, this method is often suboptimal as it forfeits an opportunity to exploit the structural characteristics typically associated with matrix analysis. To address this challenge, we propose a novel differential privacy mechanism called the Matrix-Variate Gaussian (MVG) mechanism, which adds a matrix-valued noise drawn from a matrix-variate Gaussian distribution, and we rigorously prove that the MVG mechanism preserves $(epsilon,delta)$-differential privacy. Furthermore, we introduce the concept of directional noise made possible by the design of the MVG mechanism. Directional noise allows the impact of the noise on the utility of the matrix-valued query function to be moderated. Finally, we experimentally demonstrate the performance of our mechanism using three matrix-valued queries on three privacy-sensitive datasets. We find that the MVG mechanism notably outperforms four previous state-of-the-art approaches, and provides comparable utility to the non-private baseline.
In this rejoinder, we aim to address two broad issues that cover most comments made in the discussion. First, we discuss some theoretical aspects of our work and comment on how this work might impact the theoretical foundation of privacy-preserving data analysis. Taking a practical viewpoint, we next discuss how f-differential privacy (f-DP) and Gaussian differential privacy (GDP) can make a difference in a range of applications.
We address the problem of how to obfuscate texts by removing stylistic clues which can identify authorship, whilst preserving (as much as possible) the content of the text. In this paper we combine ideas from generalised differential privacy and machine learning techniques for text processing to model privacy for text documents. We define a privacy mechanism that operates at the level of text documents represented as bags-of-words - these representations are typical in machine learning and contain sufficient information to carry out many kinds of classification tasks including topic identification and authorship attribution (of the original documents). We show that our mechanism satisfies privacy with respect to a metric for semantic similarity, thereby providing a balance between utility, defined by the semantic content of texts, with the obfuscation of stylistic clues. We demonstrate our implementation on a fan fiction dataset, confirming that it is indeed possible to disguise writing style effectively whilst preserving enough information and variation for accurate content classification tasks.
Differential Privacy protects individuals data when statistical queries are published from aggregated databases: applying obfuscating mechanisms to the query results makes the released information less specific but, unavoidably, also decreases its utility. Yet it has been shown that for discrete data (e.g. counting queries), a mandated degree of privacy and a reasonable interpretation of loss of utility, the Geometric obfuscating mechanism is optimal: it loses as little utility as possible. For continuous query results however (e.g. real numbers) the optimality result does not hold. Our contribution here is to show that optimality is regained by using the Laplace mechanism for the obfuscation. The technical apparatus involved includes the earlier discrete result by Ghosh et al., recent work on abstract channels and their geometric representation as hyper-distributions, and the dual interpretations of distance between distributions provided by the Kantorovich-Rubinstein Theorem.
Traditionally, differential privacy mechanism design has been tailored for a scalar-valued query function. Although many mechanisms such as the Laplace and Gaussian mechanisms can be extended to a matrix-valued query function by adding i.i.d. noise to each element of the matrix, this method is often sub-optimal as it forfeits an opportunity to exploit the structural characteristics typically associated with matrix analysis. In this work, we consider the design of differential privacy mechanism specifically for a matrix-valued query function. The proposed solution is to utilize a matrix-variate noise, as opposed to the traditional scalar-valued noise. Particularly, we propose a novel differential privacy mechanism called the Matrix-Variate Gaussian (MVG) mechanism, which adds a matrix-valued noise drawn from a matrix-variate Gaussian distribution. We prove that the MVG mechanism preserves $(epsilon,delta)$-differential privacy, and show that it allows the structural characteristics of the matrix-valued query function to naturally be exploited. Furthermore, due to the multi-dimensional nature of the MVG mechanism and the matrix-valued query, we introduce the concept of directional noise, which can be utilized to mitigate the impact the noise has on the utility of the query. Finally, we demonstrate the performance of the MVG mechanism and the advantages of directional noise using three matrix-valued queries on three privacy-sensitive datasets. We find that the MVG mechanism notably outperforms four previous state-of-the-art approaches, and provides comparable utility to the non-private baseline. Our work thus presents a promising prospect for both future research and implementation of differential privacy for matrix-valued query functions.
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