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

TL;DR

This paper introduces an optimal, efficient Fourier-based mechanism for answering weighted marginal and product queries under differential privacy, improving on prior methods in simplicity, speed, and optimality.

Contribution

The authors present a polynomial-time, optimal factorization mechanism for differentially private marginal and product queries using Fourier transforms, simplifying previous approaches.

Findings

Mechanism is exactly optimal among factorization methods for minimizing noise variances.

Algorithm runs in polynomial time, faster than previous SDP-based methods.

Achieves near-optimal noise variance for extended marginal queries.

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.