Optimal Schemes for Discrete Distribution Estimation under Locally Differential Privacy (1702.00610v1)

Abstract: We consider the minimax estimation problem of a discrete distribution with support size $k$ under privacy constraints. A privatization scheme is applied to each raw sample independently, and we need to estimate the distribution of the raw samples from the privatized samples. A positive number $\epsilon$ measures the privacy level of a privatization scheme. For a given $\epsilon,$ we consider the problem of constructing optimal privatization schemes with $\epsilon$-privacy level, i.e., schemes that minimize the expected estimation loss for the worst-case distribution. Two schemes in the literature provide order optimal performance in the high privacy regime where $\epsilon$ is very close to $0,$ and in the low privacy regime where $e^{\epsilon}\approx k,$ respectively. In this paper, we propose a new family of schemes which substantially improve the performance of the existing schemes in the medium privacy regime when $1\ll e^{\epsilon} \ll k.$ More concretely, we prove that when $3.8 < \epsilon <\ln(k/9) ,$ our schemes reduce the expected estimation loss by $50\%$ under $\ell_2^2$ metric and by $30\%$ under $\ell_1$ metric over the existing schemes. We also prove a lower bound for the region $e^{\epsilon} \ll k,$ which implies that our schemes are order optimal in this regime.

• The paper introduces new privatization schemes that significantly reduce estimation loss for discrete distribution estimation in the medium privacy regime.
• The proposed methods achieve a 50% reduction in ℓ² loss and a 30% reduction in ℓ₁ loss, outperforming established techniques like k-RAPPOR and k-RR.
• The authors provide rigorous lower bound proofs that confirm these schemes are order-optimal for locally differentially private estimation.

An Overview of "Optimal Schemes for Discrete Distribution Estimation under Locally Differential Privacy"

This paper by Min Ye and Alexander Barg addresses the challenge of estimating discrete distributions under locally differential privacy constraints. The authors focus on constructing optimal privatization schemes that minimize the estimation loss given a privacy parameter ε, which quantifies the level of privacy protection. Two known schemes, $k$-RAPPOR and $k$-RR, are acknowledged to perform optimally in high and low privacy regimes, respectively. The authors propose new schemes that offer improved performance in the medium privacy regime.

Key Contributions

1. New Privatization Schemes: The paper introduces a class of privatization schemes parameterized by an integer $d$, improving performance in the medium privacy regime ($1 \ll e^{\epsilon} \ll k$). The schemes reduce the expected estimation loss significantly compared to existing methods.
2. Numerical Improvements: The authors rigorously demonstrate that their proposed schemes achieve a 50% reduction in estimation loss under the $\ell^2_2$ metric and a 30% reduction under the $\ell_1$ metric when $3.8 < \epsilon < \ln(k/9)$.
3. Lower Bound Proofs: The paper includes proofs that establish these schemes as order-optimal within the studied privacy regime, providing tight lower bounds that underscore their efficiency.

Theoretical and Practical Implications

The advancement in privacy-preserving distribution estimation could significantly impact how large datasets, especially those containing sensitive information, are analyzed. Privacy constraints are becoming increasingly stringent due to regulatory considerations and the growing public consciousness around data security. These methods could facilitate more precise statistical analysis without compromising data privacy.

Future Directions

Further exploration into the trade-offs between privacy and accuracy in various data-rich environments is warranted. As the landscape of privacy-preserving data analysis evolves, understanding the theoretical limits and empirical validations of such schemes is crucial. Additionally, deploying these schemes in real-world applications to understand their computational efficiency and adaptability in dynamic settings would be valuable.

In conclusion, this paper contributes an important perspective on locally differential privacy and offers practical schemes that balance privacy and accuracy. The results and methods presented are vital for researchers working on privacy-preserving statistics and related fields, providing a foundation for subsequent analyses and applications in data privacy.