Sort, Partition, Randomize: Optimal Binary Hypothesis Testing under Local Differential Privacy

2026-06-05

AI Summary

This work addresses the design of optimal mechanisms for binary hypothesis testing under ε-local differential privacy (LDP). It proposes the Sort-Partition-Randomize (SPR) framework, which first orders input symbols by their likelihood ratios, partitions them into contiguous blocks, and then applies randomized response to the block labels. Leveraging this structure, the paper establishes the existence of an optimal mechanism for any privacy budget ε and any f-divergence–based utility objective—including total variation distance and KL divergence—and presents, for the first time, a dynamic programming algorithm that computes such a mechanism exactly in O(k³) time. This approach overcomes prior limitations restricted to asymptotic privacy regimes, enabling efficient computation of optimal mechanisms across the full range of privacy parameters.

Abstract

We study optimal design of $\varepsilon$-locally differentially private mechanisms for binary hypothesis testing. Each observation is drawn from one of two known distributions $P_0,P_1$ on a finite alphabet of size $k$, privatized by a mechanism $Q$, and then used to infer which distribution generated the data. We measure testing utility using an $f$-divergence, including total variation, KL, and hockey-stick divergences, between the two induced output distributions. Previous work established structural properties of optimal mechanisms, but only yielded exponential-time algorithms. We prove a sharp structure: for every $\varepsilon$ and every $f$-divergence objective, after sorting the alphabet by likelihood ratio, there exists an optimal mechanism that partitions the sorted alphabet into contiguous blocks and applies randomized response to the block label. We call this class Sort-Partition-Randomize (SPR). This characterization yields an exact dynamic program that computes an optimal mechanism in $O(k^3)$ time, and more generally in $O(\ell k^2)$ time with an $\ell$-output budget. Our results make it possible to efficiently compute and characterize the exact optimum across the full privacy range, beyond asymptotic privacy regimes.

Problem

• local differential privacy
• binary hypothesis testing
• optimal mechanism design
• f-divergence
• privacy-utility tradeoff

Innovation

• Local Differential Privacy
• Binary Hypothesis Testing
• Sort-Partition-Randomize
• f-divergence
• Dynamic Programming