Interaction Is Not Necessary for Order-Optimal 1-Bit Mean Estimation

This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on ℝ with mean in [-λ,λ] and absolute k-th central moment at most σk, where k>1 is fixed. For this class, previous work attained the optimal sample complexity for general queries using a two-stage protocol. The first stage localizes the mean. The second-stage queries are chosen after localization and refine the estimate around the decoded center. We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol that fixes all queries before observing the data and matches the optimal adaptive sample complexity. For target accuracy ε and confidence 1-δ, its sample complexity scales as [ logfracλσ + begincases (σ/ε)2log(1/δ), & k>2, (σ/ε)2log(σ/ε)log(1/δ), & k=2, (σ/ε)k/(k-1)log(1/δ), & 1<k<2, endcases ] up to constants depending only on k. In the range covered by the known lower bound, this rate is minimax optimal even among fully adaptive protocols. This gives a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation with general queries citep[Open Problem~1]lau2026open.