From Robustness to Privacy and Back (2302.01855v1)

Abstract: We study the relationship between two desiderata of algorithms in statistical inference and machine learning: differential privacy and robustness to adversarial data corruptions. Their conceptual similarity was first observed by Dwork and Lei (STOC 2009), who observed that private algorithms satisfy robustness, and gave a general method for converting robust algorithms to private ones. However, all general methods for transforming robust algorithms into private ones lead to suboptimal error rates. Our work gives the first black-box transformation that converts any adversarially robust algorithm into one that satisfies pure differential privacy. Moreover, we show that for any low-dimensional estimation task, applying our transformation to an optimal robust estimator results in an optimal private estimator. Thus, we conclude that for any low-dimensional task, the optimal error rate for $\varepsilon$-differentially private estimators is essentially the same as the optimal error rate for estimators that are robust to adversarially corrupting $1/\varepsilon$ training samples. We apply our transformation to obtain new optimal private estimators for several high-dimensional tasks, including Gaussian (sparse) linear regression and PCA. Finally, we present an extension of our transformation that leads to approximate differentially private algorithms whose error does not depend on the range of the output space, which is impossible under pure differential privacy.

• The paper presents a black-box transformation that converts robust algorithms into differentially private ones while preserving optimal error rates under specific dimensional and corruption conditions.
• The research establishes error rate equivalence for low-dimensional tasks, showing that robust and privacy-preserving estimators can achieve similar minimax error rates.
• The work extends the methodology to high-dimensional tasks and approximate differential privacy, offering a practical approach for designing efficient estimators with balanced performance.

From Robustness to Privacy and Back: A Formal Summary

The paper "From Robustness to Privacy and Back" establishes a novel theoretical connection between differential privacy (DP) and adversarial robustness in statistical inference and machine learning. Two important algorithmic properties—differential privacy and robustness to data corruption—have long been studied independently. This paper bridges them by developing a black-box transformation, providing a clear methodology to construct differentially private algorithms from robust ones while retaining optimal error rates.

Key Contributions

1. Black-Box Transformation: The authors introduce a transformation that converts robust algorithms to differentially private algorithms. This transformation preserves the accuracy characteristics of the original robust algorithm, provided that certain conditions on dimensionality and corruptibility parameters are met.
2. Error Rate Equivalence: For low-dimensional tasks, the transformation is proven to be optimal, meaning that the transformed algorithm's error aligns with state-of-the-art differentially private algorithms. This shows that the minimax error rates for $\epsilon$-DP estimators are essentially the same as for robust estimators that tolerate a corruption fraction inversely proportional to the privacy parameter.
3. Application to High-Dimensional Tasks: The paper extends its methodology to high-dimensional tasks, such as Gaussian linear regression, sparse linear regression, and Principal Component Analysis (PCA). By leveraging robust statistical methods, the transformation yields differentially private estimators optimized for these complex tasks.
4. Approximate Differential Privacy Extension: An extension of the transformation is presented for constructing algorithms that satisfy approximate differential privacy. Here, the error is decoupled from the range of the output space—an improvement that pure DP cannot achieve.

Theoretical Insights and Implications

• Dimensional Dependency: The transformation ensures that for low-dimensional problems, $\tau \approx \log(n)/(n\epsilon)$ suffices, where $\tau$ is the fraction of data corruption the robust algorithm can withstand, $n$ the dataset size, and $\epsilon$ the privacy parameter. For high-dimensional problems, the transformation's error rate incurs a factor proportional to the dimension, unless additional structural assumptions like sparsity are exploited.
• Implications for Algorithm Design: As robust estimators are inherently less sensitive to small input changes, this transformation underlines a pathway to design private algorithms efficiently, without starting from scratch for each estimation task.

Practical Implications and Future Directions

• Practical Algorithm Design: The proposed black-box transformation simplifies constructing differentially private algorithms for practitioners who already have robust statistical estimators at hand.
• Performance Balance: The transformation highlights the balance between robustness and privacy and opens the possibility of designing estimators that can achieve both properties simultaneously within acceptable error margins.
• Extension to Other Distributions: For future research, extending the principles demonstrated here to other probability distributions, such as heavy-tailed distributions, could broaden the applicability of this work.
• Computational Efficiency: While currently computationally intensive, future work might explore efficient instantiations of this transformation that maintain the theoretical guarantees.

Overall, the paper "From Robustness to Privacy and Back" establishes a foundational methodology that intertwines robustness and privacy in statistical estimation, demonstrating their interchangeability in error rates for low-dimensional tasks and setting the stage for more nuanced algorithm development in the field of high-dimensional data processing.