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

2026-08-03

AI Summary

This work addresses the problem of mean estimation under single-bit communication constraints, aiming to resolve the COLT 2026 open question of whether interactive protocols are necessary to achieve optimal sample complexity. We propose a fully non-adaptive randomized protocol and establish, for the first time, that interaction is not required in this setting. By leveraging non-adaptive query design, modeling via central moment constraints, and conducting a case-specific complexity analysis, our method achieves minimax-optimal sample complexity across different $k$-th moment conditions: $(\sigma/\varepsilon)^2 \log(1/\delta)$ for $k > 2$, $(\sigma/\varepsilon)^2 \log(\sigma/\varepsilon) \log(1/\delta)$ for $k = 2$, and $(\sigma/\varepsilon)^{k/(k-1)} \log(1/\delta)$ for $1 < k < 2$.

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}.

Problem

• one-bit mean estimation
• non-adaptive protocol
• sample complexity
• minimax optimality
• interaction

Innovation

• one-bit mean estimation
• non-adaptive protocol
• minimax optimality
• sample complexity
• randomized quantization