AI 中文总结
研究在有限中心矩下最优阶1比特均值估计问题，构造完全非自适应公共硬币协议，通过两种互补构造实现解码器端细化，给出不同\(k\)值下的细化成本，回答了相关开放问题，样本复杂度在特定范围内是极小极大最优的。
AI 中文摘要
本文表明，在有限中心矩下，最优阶1比特均值估计无需交互。对于满足\(|\mathbb{E}X|\leq\lambda\)且\(\mathbb{E}|X - \mathbb{E}X|^k\leq\sigma^k\)（固定\(k>1\)）的分布，我们构造了一个完全非自适应的公共硬币协议，在通信前确定每个可测1比特查询。所有定位和细化查询一次性生成，后续解码的粗中心仅改变存储的细化比特的解释方式。通过两种互补构造实现解码器端细化：基于周期余数的有限二元方案和基于移位随机网格的连续尺度方案。对于\(k>2\)，细化成本为\((\sigma/\epsilon)^2\log(1/\delta)\)；对于\(k = 2\)，为\((\sigma/\epsilon)^2[1+\log(\sigma/\epsilon)]\log(1/\delta)\)；对于\(1<k<2\)，为\((\sigma/\epsilon)^{k/(k - 1)}\log(1/\delta)\)。这些速率与加法定位成本\(1+\log(\lambda/\sigma)\)一起肯定地回答了Lau - Scarlett关于任意可测1比特查询的开放问题。在现有小误差、高置信度下界覆盖的参数范围内，所得样本复杂度是极小极大最优的。
英文摘要
This paper shows that interaction is unnecessary for order-optimal 1-bit mean estimation under finite central moments. For distributions satisfying $|\mathbb{E}X|\leqλ$ and $\mathbb{E}|X-\mathbb{E}X|^k\leqσ^k$ for a fixed $k>1$, we construct a fully non-adaptive public-coin protocol that fixes every measurable 1-bit query before communication. All localization and refinement queries are generated in a single batch; a subsequently decoded coarse center changes only how the stored refinement bits are interpreted. Two complementary constructions realize this decoder-side refinement: a finite dyadic scheme based on periodic residues and a continuous-scale scheme based on shifted random grids. Up to $k$-dependent constants, the refinement cost is $(σ/ε)^2\log(1/δ)$ for $k>2$, $(σ/ε)^2[1+\log(σ/ε)]\log(1/δ)$ for $k=2$, and $(σ/ε)^{k/(k-1)}\log(1/δ)$ for $1<k<2$. Together with the additive localization cost $1+\log(λ/σ)$, these rates answer the Lau--Scarlett open problem for arbitrary measurable 1-bit queries in the affirmative. In the parameter range covered by existing small-error, high-confidence lower bounds, the resulting sample complexity is minimax optimal.
Comments22 pages, 4 figures
