Finite Sample Differentially Private Confidence Intervals (1711.03908v1)

Abstract: We study the problem of estimating finite sample confidence intervals of the mean of a normal population under the constraint of differential privacy. We consider both the known and unknown variance cases and construct differentially private algorithms to estimate confidence intervals. Crucially, our algorithms guarantee a finite sample coverage, as opposed to an asymptotic coverage. Unlike most previous differentially private algorithms, we do not require the domain of the samples to be bounded. We also prove lower bounds on the expected size of any differentially private confidence set showing that our the parameters are optimal up to polylogarithmic factors.

• The paper develops innovative DP mechanisms that construct finite-sample confidence intervals with lengths matching non-private cases up to polylogarithmic factors.
• It introduces a novel range estimation method that eliminates the need for predefined data bounds by leveraging DP histogram learners and properties of the normal distribution.
• It establishes lower bounds confirming that the additional length for privacy is necessary, closely aligning with optimal theoretical limits in statistical estimation.

Overview of Finite Sample Differentially Private Confidence Intervals

This paper provides a comprehensive analysis of estimating finite sample confidence intervals for the mean of a normal distribution under differential privacy constraints. The work explores both scenarios where the variance is known and unknown, presenting differentially private algorithms that construct confidence intervals with finite sample coverage instead of relying solely on asymptotic coverage. An important feature of these algorithms is their ability to function without requiring the sample domain bounded, a requirement of most preceding differential privacy algorithms.

Key Contributions and Results

1. Algorithm Development:
   • The paper introduces differentially private mechanisms for estimating confidence intervals of a normal mean with known and unknown variance, ensuring $\epsilon$-differential privacy for all input datasets.
   • For known variance, the algorithm guarantees that with sufficient sample size, the confidence interval reaches optimal length up to polylogarithmic factors. The expected length of the interval is a maximum of terms $\frac{\sigma}{\sqrt{n}} \sqrt{\log(1/\alpha)}$, matching non-private cases, and vanishes linearly in $n$ as $\frac{\sigma}{\epsilon n} \text{polylog}(n/\alpha)$.
   • In the unknown variance scenario, a similarly structured mechanism provides confidence intervals with expected lengths having comparable bounds, demonstrating nearly optimal performance.
2. Range Estimation:
   • The paper addresses a common challenge by estimating data range differentially, eliminating the need for predetermined data bounds. This is achieved by leveraging properties of normal distributions and employing differentially private histogram learners.
   • The mechanism estimates the range such that all data points lie within a certain interval with high probability, crucially without biasing the interval estimation.
3. Lower Bounds:
   • Demonstrates that the proposed confidence interval lengths approach the theoretical lower bounds for differential privacy. Specifically, it establishes that $\Omega(\sigma / \epsilon n) \log(1/\alpha)$ is a necessary cost of privacy in interval length, thereby validating the achieved bounds as close to optimal.

Implications and Prospects

The algorithms designed in this paper are significant for the development of privacy-preserving statistical inference mechanisms that maintain statistical validity in finite samples. Procuring valid and tight confidence intervals without bounded input assumptions can substantially enhance the applicability of differential privacy in real-world data analysis, especially in domains such as social sciences and healthcare, where stringent privacy protection is paramount.

The work furthers the melding of conservative statistical inference with rigorous privacy protection, which remains critical in deploying privacy-preserving techniques that withstand varied sample distributions and sizes. These contributions pave the way for extending differentially private methods to other common statistical models and encourage investigation into other forms of inference procedures beyond interval estimation.

Future Directions

Closing the gap between current differentially private and classical non-private statistical methods remains an open challenge. Specifically, exploring whether intervals can match non-private lengths with large samples is an intriguing perspective, potentially recalibrating the balance between privacy cost and inferential precision.

Moreover, practical applications of these methodologies invite further exploration of variants optimized for computational efficiency and real-world datasets. Such endeavors can enhance the viability and adoption of differential privacy in common statistical practices, embedding privacy-preserving techniques more deeply within the analytical fabric of large data ecosystems.

This work's comprehensive approach serves as a solid foundation for the ongoing effort to merge theoretical privacy guarantees with practical statistical requirements, ensuring that sensitive data can be analyzed without jeopardizing privacy—an ever-critical endeavor in the data-driven age.