N-output Mechanism: Estimating Statistical Information from Numerical Data under Local Differential Privacy

2025-10-13

AI Summary

Existing local differential privacy (LDP) mechanisms for numeric data lack a unified optimization framework for arbitrary finite output cardinality (N); optimal perturbation schemes are known only for degenerate cases where the output space size (|mathcal{Y}|) is either extremely small ((2) or (3)) or infinite. Method: We propose the first general-purpose LDP mechanism adaptable to any discrete output size (N), derived by jointly optimizing the minimum-variance unbiased estimation problem under LDP constraints. Our approach integrates closed-form analytical derivation with efficient numerical optimization and naturally extends to mean, variance, and distribution estimation. Contribution/Results: The mechanism achieves Pareto-optimal trade-offs between estimation accuracy and privacy. Experiments demonstrate state-of-the-art accuracy across multiple statistical estimation tasks, with low communication overhead and significant improvements over existing LDP baselines.

Technology Category

Machine Learning: PrivacySearch and Optimization: Mixed Discrete/Continuous SearchReasoning under Uncertainty: Stochastic Optimization

Application Category

Security and Privacy: Large-scale security measurementsUser Modeling, Personalization and Recommendation: User privacy protection in personalized systemsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Local Differential Privacy (LDP) addresses significant privacy concerns in sensitive data collection. In this work, we focus on numerical data collection under LDP, targeting a significant gap in the literature: existing LDP mechanisms are optimized for either a very small ($|Ω| in {2, 3}$) or infinite output spaces. However, no generalized method for constructing an optimal mechanism for an arbitrary output size $N$ exists. To fill this gap, we propose the extbf{N-output mechanism}, a generalized framework that maps numerical data to one of $N$ discrete outputs. We formulate the mechanism's design as an optimization problem to minimize estimation variance for any given $N geq 2$ and develop both numerical and analytical solutions. This results in a mechanism that is highly accurate and adaptive, as its design is determined by solving an optimization problem for any chosen $N$. Furthermore, we extend our framework and existing mechanisms to the task of distribution estimation. Empirical evaluations show that the N-output mechanism achieves state-of-the-art accuracy for mean, variance, and distribution estimation with small communication costs.

Problem

• Optimizing LDP mechanisms for arbitrary discrete output sizes
• Minimizing estimation variance in numerical data collection
• Extending LDP framework for accurate distribution estimation tasks

Innovation

• Generalized N-output mechanism for numerical data
• Optimization-based design minimizing estimation variance
• Extends to distribution estimation with small communication