Local, Private, Efficient Protocols for Succinct Histograms (1504.04686v1)

Abstract: We give efficient protocols and matching accuracy lower bounds for frequency estimation in the local model for differential privacy. In this model, individual users randomize their data themselves, sending differentially private reports to an untrusted server that aggregates them. We study protocols that produce a succinct histogram representation of the data. A succinct histogram is a list of the most frequent items in the data (often called "heavy hitters") along with estimates of their frequencies; the frequency of all other items is implicitly estimated as 0. If there are $n$ users whose items come from a universe of size $d$, our protocols run in time polynomial in $n$ and $\log(d)$. With high probability, they estimate the accuracy of every item up to error $O\left(\sqrt{\log(d)/(\epsilon^2n)}\right)$ where $\epsilon$ is the privacy parameter. Moreover, we show that this much error is necessary, regardless of computational efficiency, and even for the simple setting where only one item appears with significant frequency in the data set. Previous protocols (Mishra and Sandler, 2006; Hsu, Khanna and Roth, 2012) for this task either ran in time $\Omega(d)$ or had much worse error (about $\sqrt[6]{\log(d)/(\epsilon^2n)}$), and the only known lower bound on error was $\Omega(1/\sqrt{n})$. We also adapt a result of McGregor et al (2010) to the local setting. In a model with public coins, we show that each user need only send 1 bit to the server. For all known local protocols (including ours), the transformation preserves computational efficiency.

• The paper introduces efficient local differential privacy protocols for succinct histograms with optimal error bounds.
• It achieves computation in polynomial time with 1-bit communication per user, reducing overall overhead.
• Lower bounds derived in the study confirm the optimal trade-off between privacy (ε) and accuracy in frequency estimation.

Analysis of "Local, Private, Efficient Protocols for Succinct Histograms"

The paper "Local, Private, Efficient Protocols for Succinct Histograms" by Raef Bassily and Adam Smith addresses a crucial problem in the field of privacy-preserving data analysis: how to efficiently construct succinct histograms while ensuring differential privacy in a local setting. This work provides innovative solutions and establishes theoretical bounds that contribute meaningfully to both the practical and academic landscapes of privacy-preserving computation.

Summary of Contributions

The authors focus on frequency estimation and identifying heavy hitters from data held by individual users, who keep their data locally private by means of randomized algorithms. Their work is situated in the local model of differential privacy, a standard approach where each user privatizes their data before sending it to an aggregator.

Key contributions of the paper are outlined as follows:

1. Efficient Protocols: The authors introduce protocols that, for the first time, enable succinct histogram generation with efficiency. The running time of these protocols is polynomial with respect to the number of users $n$ and the logarithm of the size of the data universe $d$. This marks a significant improvement over earlier methods that had higher computational requirements or less optimal accuracy bounds.
2. Optimal Accuracy Bounds: The proposed protocols achieve an accuracy of $O(\sqrt{\log(d)/(\epsilon^2 n)})$, which is proven to be optimal for the problem under consideration. This result is pivotal because it establishes the best possible trade-off between privacy (parameterized by $\epsilon$) and accuracy, a balance that is essential for practical deployment.
3. Lower Bounds: Through theoretical analysis, the authors derive lower bounds on the error for any local differential privacy protocol in this setting. They show that the error must be at least $\Omega(\sqrt{\log(d)/(\epsilon^2 n)})$ for frequency estimation, even if one item dominates the data set. This insight underscores the fundamental limits of privacy-preserving algorithms, regardless of computational resources available.
4. 1-bit Protocols: By adapting techniques from compressive sensing and streaming algorithms, the authors show how each user's communication can be reduced to a single bit without compromising the accuracy or efficiency. This result is particularly compelling for applications where communication overhead is critical.

Implications and Speculations

The implications of this work are substantial for the field of differential privacy. Practically, the ability to compute succinct histograms efficiently while maintaining robust privacy guarantees expands the horizon for privacy-preserving methodologies in real-world applications, such as data analytics in web browsers or financial systems.

Theoretically, the tight bounds on error provide a benchmark against which future local differential privacy algorithms may be measured. The fusion of coding theory, hashing mechanisms, and differential privacy in this work could inspire similar interdisciplinary approaches to other complex privacy-preserving tasks.

Considering future developments, one could anticipate further exploration of the techniques presented in this paper for other data analytic tasks beyond histogram computation. Moreover, these methods might be extended to more complex data models or enriched to handle richer queries while maintaining local privacy.

Conclusion

Overall, the paper makes significant strides in addressing the complexities of data privacy in distributed environments. The authors' methodical approach to establishing both upper and lower error bounds, coupled with computationally efficient solutions, marks a noteworthy advancement in the design and understanding of local differential privacy mechanisms. As data privacy continues to be a critically important concern, the insights from this paper will likely influence both ongoing research and practical applications in the field.