Simple Binary Hypothesis Testing Under Local Differential Privacy and Communication Constraints

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {A}$ </tex-math></inline-formula> of two distributions <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i>, defined as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {A}:= \{(\vec {T} p, \vec {T} q) | \vec {T}\in \mathcal {C}\}$ </tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> is the set of channels characterizing the constraints.