发表机构
Shanghai University of Finance and Economics; Shanghai Jiao Tong University（上海财经大学; 上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对1比特均值估计问题，提出一种随机化完全非自适应协议，无需交互即可达到与自适应协议一致的极小极大最优样本复杂度，否定了COLT 2026相关开放问题。
AI 中文摘要
本文研究1比特均值估计问题，其中每个独立样本由单条二进制消息表示。我们考虑定义在实数域$\u211d$上的分布，其均值落在$[-\u03bb,\u03bb]$区间内，且绝对$k$阶中心矩不超过$\u03c3^k$，其中$k>1$为固定值。针对这类分布，已有研究采用两阶段协议实现了通用查询下的最优样本复杂度：第一阶段对均值进行定位；第二阶段在定位完成后选择查询，围绕解码得到的中心值细化估计结果。我们证明这种交互是可以避免的：我们构造了一种随机化的完全非自适应协议，该协议在观测数据之前就固定所有查询，且能达到与自适应协议相同的最优样本复杂度。对于目标精度$\u03b5$和置信度$1-\u03b4$，其样本复杂度的缩放形式为$\u03bb/\u03c3$的对数加上分段函数：当$k>2$时为$(\u03c3/\u03b5)^2\uf06c\u03bf\u03b3(1/\u03b4)$；当$k=2$时为$(\u03c3/\u03b5)^2\uf06c\u03bf\u03b3(\u03c3/\u03b5)\uf06c\u03bf\u03b3(1/\u03b4)$；当$1<k<2$时为$(\u03c3/\u03b5)^{k/(k-1)}\uf06c\u03bf\u03b3(1/\u03b4)$，误差仅取决于$k$的常数因子。在已知下界覆盖的范围内，该速率即使在完全自适应协议中也属于极小极大最优。这对COLT 2026的开放问题给出了否定答案，该问题询问在通用查询的阶最优1比特均值估计中交互是否是必需的。
英文摘要
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}.
