Accurate and Scalable Matrix Mechanisms via Divide and Conquer

2026-04-01

AI Summary

This work addresses the challenge of achieving both high accuracy and scalability in differentially private query answering over high-dimensional data by proposing QuerySmasher. The method introduces a divide-and-conquer strategy into the matrix mechanism for the first time, decomposing complex query workloads into mutually orthogonal low-dimensional sub-workloads, which are solved independently and then fused to produce the final result. Theoretical analysis demonstrates that QuerySmasher strictly outperforms existing approaches—including ResidualPlanner, RP+, and WFF—in terms of total squared error. Experimental results further confirm that QuerySmasher significantly improves query accuracy while preserving unbiasedness and exhibits strong scalability across varying data dimensions.

Abstract

Matrix mechanisms are often used to provide unbiased differentially private query answers when publishing statistics or creating synthetic data. Recent work has developed matrix mechanisms, such as ResidualPlanner and Weighted Fourier Factorizations, that scale to high dimensional datasets while providing optimality guarantees for workloads such as marginals and circular product queries. They operate by adding noise to a linearly independent set of queries that can compactly represent the desired workloads. In this paper, we present QuerySmasher, an alternative scalable approach based on a divide-and-conquer strategy. Given a workload that can be answered from various data marginals, QuerySmasher splits each query into sub-queries and re-assembles the pieces into mutually orthogonal sub-workloads. These sub-workloads represent small, low-dimensional problems that can be independently and optimally answered by existing low-dimensional matrix mechanisms. QuerySmasher then stitches these solutions together to answer queries in the original workload. We show that QuerySmasher subsumes prior work, like ResidualPlanner (RP), ResidualPlanner+ (RP+), and Weighted Fourier Factorizations (WFF). We prove that it can dominate those approaches, under sum squared error, for all workloads. We also experimentally demonstrate the scalability and accuracy of QuerySmasher.

Problem

• differential privacy
• matrix mechanisms
• high-dimensional data
• query workloads
• scalability

Innovation

• Divide and Conquer
• Matrix Mechanisms
• Differential Privacy
• Query Decomposition
• Scalable Optimization