Weighted Fourier Factorizations: Optimal Gaussian Noise for Differentially Private Marginal and Product Queries

2025-12-24

AI Summary

This paper addresses the optimal noise addition problem for weighted marginal and product queries under differential privacy, aiming to minimize either the weighted average or the maximum noise variance. We propose a Fourier-domain noise allocation mechanism: independent Gaussian noise with query-specific variances is added in the Fourier basis, followed by inverse discrete Fourier transform to reconstruct the query answers. Our approach yields the **exact optimal** noise allocation—minimizing either the weighted average or the maximum variance—for arbitrary weighted marginal and product queries, and provides low-order approximate optimality for extended marginal queries. The mechanism is analytically tractable and provably optimal; its time complexity is polynomial in both input and output sizes, offering substantial improvement over existing semidefinite programming (SDP)-based methods.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We revisit the task of releasing marginal queries under differential privacy with additive (correlated) Gaussian noise. We first give a construction for answering arbitrary workloads of weighted marginal queries, over arbitrary domains. Our technique is based on releasing queries in the Fourier basis with independent noise with carefully calibrated variances, and reconstructing the marginal query answers using the inverse Fourier transform. We show that our algorithm, which is a factorization mechanism, is exactly optimal among all factorization mechanisms, both for minimizing the sum of weighted noise variances, and for minimizing the maximum noise variance. Unlike algorithms based on optimizing over all factorization mechanisms via semidefinite programming, our mechanism runs in time polynomial in the dataset and the output size. This construction recovers results of Xiao et al. [Neurips 2023] with a simpler algorithm and optimality proof, and a better running time. We then extend our approach to a generalization of marginals which we refer to as product queries. We show that our algorithm is still exactly optimal for this more general class of queries. Finally, we show how to embed extended marginal queries, which allow using a threshold predicate on numerical attributes, into product queries. We show that our mechanism is almost optimal among all factorization mechanisms for extended marginals, in the sense that it achieves the optimal (maximum or average) noise variance up to lower order terms.

Problem

• Releasing marginal queries with optimal Gaussian noise under differential privacy.
• Extending optimal noise methods to product queries and extended marginals.
• Achieving polynomial runtime while maintaining exact optimality for factorization mechanisms.

Innovation

• Fourier basis noise calibration for marginals
• Optimal factorization mechanism for product queries
• Polynomial time algorithm with optimal noise variance