We study simple binary hypothesis testing under both local differential privacy (LDP) and communication constraints. We qualify our results as either minimax optimal or instance optimal: the former hold for the set of distribution pairs with prescribed Hellinger divergence and total variation distance, whereas the latter hold for specific distribution pairs. For the sample complexity of simple hypothesis testing under pure LDP constraints, we establish instance-optimal bounds for distributions with binary support; minimax-optimal bounds for general distributions; and (approximately) instance-optimal, computationally efficient algorithms for general distributions. When both privacy and communication constraints are present, we develop instance-optimal, computationally efficient algorithms that achieve the minimum possible sample complexity (up to universal constants). Our results on instance-optimal algorithms hinge on identifying the extreme points of the joint range set $\mathcal A$ of two distributions $p$ and $q$, defined as $\mathcal A := \{(\mathbf T p, \mathbf T q) | \mathbf T \in \mathcal C\}$, where $\mathcal C$ is the set of channels characterizing the constraints.

翻译：我们研究了在局部差分隐私（LDP）和通信约束共同作用下的简单二元假设检验问题。我们的结果可分为极小极大最优或实例最优两类：前者适用于具有指定Hellinger散度和总变差距离的分布对集合，而后者则针对特定分布对。针对纯LDP约束下简单假设检验的样本复杂度，我们建立了：二元支撑分布族的实例最优下界；一般分布族的极小极大最优下界；以及(近似)实例最优且计算高效的一般分布族算法。当隐私与通信约束同时存在时，我们提出了实例最优且计算高效的算法，该算法能达到最小可能的样本复杂度（至多相差通用常数）。实例最优算法的核心在于识别两个分布$p$和$q$的联合值域集合$\mathcal A$的极值点，其中$\mathcal A := \{(\mathbf T p, \mathbf T q) | \mathbf T \in \mathcal C\}$，而$\mathcal C$为刻画约束的通道集合。