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

• URL: http://arxiv.org/abs/2608.02538v1
• Date: Mon, 03 Aug 2026 17:28:12 GMT
• Title: Interaction Is Not Necessary for Order-Optimal 1-Bit Mean Estimation
• Abstract summary: We consider distributions on $mathbbR$ with mean in $[-,]$ and absolute $k$-th central moment at most $k$, where $k>1$ is fixed.<n>We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol.
• Score: 14.106326180143666
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on $\mathbb{R}$ with mean in $[-λ,λ]$ and absolute $k$-th central moment at most $σ^k$, 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 \[ \log\fracλσ + \begin{cases} (σ/ε)^2\log(1/δ), & k>2,\\ (σ/ε)^2\log(σ/ε)\log(1/δ), & k=2,\\ (σ/ε)^{k/(k-1)}\log(1/δ), & 1<k<2, \end{cases} \] 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}.

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