Exploiting Coordinate-wise Disparity to Achieve Higher Utility in Differentially Private Mechanisms

Core Concepts

By adding independent but non-identically distributed (i.n.i.d.) noise across the coordinates of a multi-dimensional query response, the proposed mechanisms can achieve higher accuracy than the conventional independent and identically distributed (i.i.d.) noise addition, while still satisfying the given differential privacy constraints.

Abstract

The article introduces the concept of adding i.n.i.d. noise to the coordinates of a multi-dimensional query response in order to achieve a tighter privacy-accuracy trade-off compared to the conventional i.i.d. noise addition.
Key highlights:

The authors formally present the i.n.i.d. Gaussian and Laplace mechanisms and derive the necessary and sufficient conditions for them to satisfy (ε, δ)-differential privacy and ε-differential privacy, respectively.
Optimal coordinate-wise scale parameters for the i.n.i.d. noise are derived, which leverage the disparity in the coordinate-wise sensitivities of the query.
Theoretical analysis and numerical simulations demonstrate that the i.n.i.d. mechanisms achieve higher utility (lower mean squared error) than their i.i.d. counterparts for the same privacy guarantees.
One key observation is that the i.n.i.d. Laplace mechanism can outperform the i.i.d. Gaussian mechanism in high dimensions, contrary to the popular belief, by exploiting the irregularity in coordinate-wise sensitivities.
The authors also show how the i.n.i.d. noise can improve the performance of private empirical risk minimization through coordinate descent.

Stats

The ℓ1-sensitivity of the K-dimensional query is denoted as ∆1.
The ℓ2-sensitivity of the K-dimensional query is denoted as ∆2.

Quotes

"One of the interesting observations is that the Laplace mechanism outperforms Gaussian even in high dimensions, as opposed to the popular belief, if the irregularity in coordinate-wise sensitivities is exploited."
"We also demonstrate how the i.n.i.d. noise can improve the performance in the private empirical risk minimization through coordinate descent."

Key Insights Distilled From

Differential Privacy with Higher Utility by Exploiting Coordinate-wise Disparity

by Gokularam Mu... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2302.03511.pdf

Deeper Inquiries

How can the proposed i.n.i.d. noise addition mechanisms be extended to other noise distributions beyond Gaussian and Laplace

The proposed i.n.i.d. noise addition mechanisms can be extended to other noise distributions beyond Gaussian and Laplace by considering the specific characteristics of the new noise distribution. The key idea is to leverage the knowledge of coordinate-wise sensitivities to determine the optimal scale parameters for the noise distribution. By understanding how the sensitivity profile of the query varies across different dimensions, it is possible to design i.n.i.d. noise mechanisms using alternative distributions that can provide better utility while ensuring differential privacy.
For example, one approach could be to explore the use of asymmetric noise distributions such as asymmetric Laplace distributions or skew distributions. These distributions can capture the non-uniformity in the sensitivities of the query responses and offer a more tailored approach to adding noise for differential privacy. By analyzing the disparities in sensitivity profiles and adapting the noise distribution accordingly, it is possible to enhance the utility of the mechanism while maintaining the desired level of privacy.

What are the potential challenges in characterizing the optimal noise distribution for multi-dimensional queries with heterogeneous coordinate-wise sensitivities

Characterizing the optimal noise distribution for multi-dimensional queries with heterogeneous coordinate-wise sensitivities can pose several challenges. One of the main challenges is the complexity of determining the interplay between the sensitivities of different dimensions and how they affect the overall utility of the mechanism. The optimal noise distribution needs to strike a balance between perturbing the query responses sufficiently to ensure privacy while minimizing the impact on the accuracy of the results.
Additionally, the computational complexity of optimizing the scale parameters for non-identically distributed noise across multiple dimensions can be a challenge. The optimization process may require sophisticated algorithms to find the optimal solution efficiently, especially when dealing with high-dimensional queries with varying sensitivities.
Another challenge is the generalizability of the optimal noise distribution across different types of queries and datasets. The characteristics of the data and the nature of the queries can influence the effectiveness of the noise mechanism, making it challenging to devise a one-size-fits-all solution for optimal noise distribution.

Can the insights from this work be leveraged to design differentially private algorithms for other machine learning tasks beyond empirical risk minimization

The insights from this work can be leveraged to design differentially private algorithms for other machine learning tasks beyond empirical risk minimization by customizing the noise addition mechanisms based on the specific requirements of the task. For example:

Clustering: In clustering algorithms, where the sensitivity of cluster assignments may vary across different clusters, the i.n.i.d. noise addition mechanisms can be tailored to account for this heterogeneity. By incorporating the disparities in sensitivity profiles, the clustering results can be privatized more effectively while preserving the clustering structure.

Principal Component Analysis (PCA): For PCA tasks, where the principal components may have different sensitivities, the optimal noise distribution can be adjusted to reflect these variations. This can help in preserving the principal components' information while ensuring privacy in the PCA results.

Linear Regression: In linear regression tasks, where the coefficients of the regression model may have diverse sensitivities, the noise addition mechanisms can be optimized to address the disparity in sensitivities. By customizing the noise distribution based on the sensitivity profile of the regression coefficients, differential privacy can be achieved without compromising the accuracy of the regression model.

By applying the principles of i.n.i.d. noise addition and considering the heterogeneous sensitivities in different machine learning tasks, it is possible to design more effective differentially private algorithms that cater to a wide range of applications beyond empirical risk minimization.