Discrete Distribution Estimation under Local Privacy (1602.07387v3)

Abstract: The collection and analysis of user data drives improvements in the app and web ecosystems, but comes with risks to privacy. This paper examines discrete distribution estimation under local privacy, a setting wherein service providers can learn the distribution of a categorical statistic of interest without collecting the underlying data. We present new mechanisms, including hashed K-ary Randomized Response (KRR), that empirically meet or exceed the utility of existing mechanisms at all privacy levels. New theoretical results demonstrate the order-optimality of KRR and the existing RAPPOR mechanism at different privacy regimes.

• The paper proves that Warner’s Randomized Response is globally optimal for binary alphabets under any loss function and privacy level.
• The study shows that while Rappor is order-optimal in high privacy settings, a hashed k-ary Randomized Response mechanism achieves lower error bounds in low privacy regimes.
• Large-scale simulations validate a projected estimator and extend the approach to open alphabets using the O-RR mechanism, effectively balancing privacy and utility.

Discrete Distribution Estimation under Local Privacy

This paper investigates the problem of estimating discrete distributions while ensuring local differential privacy, an important topic due to the increasing concerns over user data privacy in the digital age. The authors introduce new mechanisms for discrete distribution estimation that outperform existing methods like Rappor, particularly focusing on the $k$-ary Randomized Response ($) mechanism and its hashed variant, O-RR.</p> <h3 class='paper-heading' id='key-contributions'>Key Contributions</h3> <ol> <li><strong>Binary Alphabets:</strong> <ul> <li>The paper proves that Warner&#39;s Randomized Response (W-RR) model is globally optimal for binary alphabets across any loss function and privacy level. This reinforces the utility of W-RR in privacy-preserving data collection and suggests that it should be favored in applications requiring binary data classification under stringent privacy constraints.</li> </ul></li> <li><strong>$k$-ary Alphabets:</strong> <ul> <li>For$k$$-ary alphabets, the study demonstrates that the Rappor mechanism is order-optimal in high privacy regimes but suboptimal in low privacy scenarios. Conversely, the$$ mechanism introduces lower error bounds and is optimal in low privacy settings. This result supports the need for tailored privacy mechanisms based on specific privacy-utility tradeoffs inherent to different privacy regimes.

Large-Scale Simulations:

• Simulations confirm that both $ and Rappor&#39;s optimal decoding depend on the true distribution&#39;s shape. The introduction of a projected estimator, effective across various privacy levels and sample sizes, showcases an improvement in decoding strategies when dealing with skewed distributions.</li> </ul></li> <li><strong>Extensions to Open Alphabets:</strong> <ul> <li>The extension to open alphabets using the O-RR mechanism represents a significant advancement, facilitating the application of <a href="https://www.emergentmind.com/topics/differential-privacy-dp" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">differential privacy</a> to scenarios where input symbols are not known a priori. Hash functions and cohort-style mechanisms allow O-RR to achieve or surpass Rappor&#39;s performance across a spectrum of privacy settings.</li> </ul></li> </ol> <h3 class='paper-heading' id='theoretical-and-empirical-analysis'>Theoretical and Empirical Analysis</h3> <ul> <li>Theoretical analyses highlight that the effective sample size needed for discrete distribution estimation is reduced when applying local differential privacy. This indicates a fundamental tradeoff between privacy and utility, where higher privacy levels necessitate larger datasets for equivalent utility.</li> <li>Empirical results from simulations further bolster the theoretical findings, showing that the hashed $k$-ary Randomized Response ($) mechanisms provide substantial utility improvements over unmodified methods and perform robustly in various settings.

Implications and Future Work

The developments in this paper have significant implications for privacy-preserving machine learning and data analysis. They suggest that appropriately tuned private mechanisms can effectively bridge the gap between utility and privacy, allowing for statistical insights without compromising user data security. Future research could explore the application of these methods to dynamic distributions and varying privacy requirements. Moreover, further investigation into domain-specific adaptations of these mechanisms remains open, particularly tailoring solutions for diverse application areas such as medical data analysis or financial transaction security.

The results clearly indicate a path toward widespread adoption of locally private techniques in industry and academia, fostering a data-centric ecosystem where privacy is a foundational principle rather than an afterthought.