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.

翻译：我们研究用于二元假设检验的$\varepsilon$-局部差分隐私机制的最优设计。每个观测值独立同分布于有限字符集（大小为$k$）上的两个已知分布$P_0,P_1$之一，经机制$Q$私有化处理后，用于推断生成该数据的分布。我们利用$f$-散度（包括全变差散度、KL散度及冰球棒散度等）衡量两个诱导输出分布之间的检验效用。先前研究已刻画最优机制的结构性质，但仅能提供指数时间算法。我们证明了一个简洁结构：对于任意$\varepsilon$和任意$f$-散度目标，在按似然比对字符集排序后，存在一个最优机制可将排序后的字符集划分为连续区块，并对区块标签应用随机化响应。我们将这类机制称为排序-划分-随机化（SPR）机制。这一刻画导出一个精确动态规划算法，可在$O(k^3)$时间内计算最优机制，更一般地，在输出预算为$\ell$的情况下可在$O(\ell k^2)$时间内完成。我们的研究结果使得在全隐私参数范围内（超越渐近隐私体制）高效计算和刻画精确最优机制成为可能。