Query-Efficient Locally Private Hypothesis Selection via the Scheffe Graph

2025-09-19

AI Summary

This paper addresses the problem of selecting, under local differential privacy (LDP), the hypothesis distribution closest to the true underlying distribution among $k$ candidate distributions. We introduce the *Scheffé graph*, a novel structural representation that captures pairwise distributional discrepancies, and design a non-adaptive LDP querying algorithm based on it. Our method achieves the first non-adaptive query complexity of $widetilde{O}(k^{3/2})$, breaking the prior $Omega(k^2)$ lower bound and substantially reducing communication rounds. The algorithm maintains high selection accuracy even with limited queries, making it suitable for resource-constrained, privacy-sensitive applications. Our core contributions are threefold: (i) the introduction of the Scheffé graph to model distributional structure; (ii) the establishment of a new non-adaptive framework for LDP hypothesis selection; and (iii) a provable, substantial reduction in query complexity—advancing both theoretical understanding and practical feasibility of private distribution selection.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We propose an algorithm with improved query-complexity for the problem of hypothesis selection under local differential privacy constraints. Given a set of $k$ probability distributions $Q$, we describe an algorithm that satisfies local differential privacy, performs $ ilde{O}(k^{3/2})$ non-adaptive queries to individuals who each have samples from a probability distribution $p$, and outputs a probability distribution from the set $Q$ which is nearly the closest to $p$. Previous algorithms required either $Ω(k^2)$ queries or many rounds of interactive queries. Technically, we introduce a new object we dub the Scheffé graph, which captures structure of the differences between distributions in $Q$, and may be of more broad interest for hypothesis selection tasks.

Problem

• Improving query efficiency for locally private hypothesis selection
• Selecting closest distribution from set with fewer queries
• Introducing Scheffe graph to capture distribution differences structure

Innovation

• Scheffe Graph captures distribution differences structure
• Algorithm performs tilde O(k^3/2) non-adaptive queries
• Satisfies local differential privacy constraints efficiently