Robust and Differentially Private Mean Estimation (2102.09159v2)

Abstract: In statistical learning and analysis from shared data, which is increasingly widely adopted in platforms such as federated learning and meta-learning, there are two major concerns: privacy and robustness. Each participating individual should be able to contribute without the fear of leaking one's sensitive information. At the same time, the system should be robust in the presence of malicious participants inserting corrupted data. Recent algorithmic advances in learning from shared data focus on either one of these threats, leaving the system vulnerable to the other. We bridge this gap for the canonical problem of estimating the mean from i.i.d. samples. We introduce PRIME, which is the first efficient algorithm that achieves both privacy and robustness for a wide range of distributions. We further complement this result with a novel exponential time algorithm that improves the sample complexity of PRIME, achieving a near-optimal guarantee and matching a known lower bound for (non-robust) private mean estimation. This proves that there is no extra statistical cost to simultaneously guaranteeing privacy and robustness.

• The paper introduces PRIME, a novel algorithm that ensures robust mean estimation with differential privacy even with an α-fraction of corrupted data.
• It employs a multiple directions filtering approach via Matrix Multiplicative Weights, significantly reducing iteration complexity compared to past methods.
• Empirical results in synthetic experiments confirm its practical viability in privacy-sensitive settings like federated learning.

An Examination of Robust and Differentially Private Mean Estimation

The paper "Robust and Differentially Private Mean Estimation" by Liu et al. addresses a significant challenge in statistical learning: the need to protect individual privacy while ensuring robustness in the face of data corruption. The work specifically focuses on mean estimation, a fundamental problem in statistics, and proposes algorithms to achieve both differential privacy (DP) and robustness against adversarial attacks. This essay provides an expert-level overview of the methodologies and contributions presented in the paper.

The authors introduce PRIME (PRIvate and robust Mean Estimation), which is an efficient algorithm to estimate the mean while ensuring both robustness and privacy. The paper tackles the dual threats of information leakage and data poisoning, common in real-world scenarios like federated learning, where data is accumulated from various sources that could be potentially malicious.

Key Contributions

1. Algorithm Design: PRIME

PRIME is designed to estimate the mean of a distribution while maintaining robustness to an $\alpha$-fraction of corrupted data and ensuring $(\varepsilon,\delta)$-DP. For computational efficiency, PRIME employs a novel multiple directions filtering approach using Matrix Multiplicative Weights, reducing the iteration complexity compared to traditional private filtering algorithms.

1. Theoretical Guarantees

The paper provides a robust theoretical foundation, showing that PRIME achieves high accuracy in mean estimation with a sample complexity of $$\widetilde{\Omega}(d/\alpha^2 + d^{3/2}/(\varepsilon\alpha)\log(1/\delta))$$. This is a substantial improvement over previous methods, which lacked robustness against adversarial corruption.

1. Exponential Time Algorithm

For scenarios where computational resources allow, the authors propose an exponential time algorithm that achieves near-optimal sample complexity for sub-Gaussian and bounded covariance distributions. This algorithm leverages the resilience properties of well-behaved distributions to ensure both robustness and differential privacy.

1. Empirical Validation

The authors empirically demonstrate the effectiveness of PRIME in synthetic settings, highlighting its robustness compared to existing DP mean estimators. The results underscore PRIME's sensitivity to both privacy requirements and dimensionality, emphasizing its practical applicability.

Methodological Innovations

• Adaptive Filtering via DP Thresholds: PRIME uses an innovative adaptive filtering mechanism, driven by the Matrix Multiplicative Weights method, which enables the simultaneous filtering of multiple directions in data. This approach significantly reduces the number of necessary iterations, enhancing both robustness and computational efficiency.
• Resilience Property for Robustness: By incorporating the resilience property into the estimation process, the authors provide a mechanism to mitigate the impact of adversarial corruption on mean estimation, thus ensuring the robustness of the DP guarantees.

Implications and Future Directions

The implications of this research are twofold. Practically, it provides a robust method for mean estimation in privacy-sensitive applications, such as those in federated and distributed learning frameworks. Theoretically, it establishes a new benchmark for the sample complexity required to achieve both robustness and differential privacy, extending the field of feasible applications for DP solutions to more adversarial settings.

Looking forward, this research opens several avenues for further exploration. One potential direction is reducing the $d^{1/2}$ factor in sample complexity, which remains an open question. Another direction could involve extending these robust DP methods to other statistical tasks, such as covariance estimation or clustering.

In conclusion, the paper by Liu et al. makes substantial contributions to the fields of privacy-preserving machine learning and robust statistics by introducing a novel algorithm that meets the dual demands of differential privacy and robustness. This work not only advances mean estimation techniques but also establishes a foundation for future research in robust and private data analysis.