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 $\tilde{O}(k^{3/2})$ 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.

翻译：我们提出了一种在本地差分隐私约束下进行假设选择问题的改进查询复杂度算法。给定一组包含$k$个概率分布$Q$的集合，我们描述了一种满足本地差分隐私的算法。该算法对每个从概率分布$p$中抽取样本的个体进行$\tilde{O}(k^{3/2})$次非自适应查询，并从集合$Q$中输出一个与$p$近似最接近的概率分布。现有算法需要$Ω(k^2)$次查询或多轮交互式查询。在技术层面，我们引入了一种称为Scheffé图的新结构，该结构能够捕捉$Q$中分布间差异的几何特性，这一工具可能对更广泛的假设选择任务具有重要研究价值。