发表机构
National University of Singapore（新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明非自适应1比特协议可达到自适应极小极大速率，并刻画了区间受限时的样本复杂度权衡，给出了保持最优速率所需的最小区间预算。
AI 中文摘要
我们研究在1比特通信约束下的分布式一维均值估计问题。每个智能体观测一个独立同分布样本，该样本来自一个未知分布，并针对中心学习器选择的查询 $Q: \mathbb{R}\to\{0,1\}$ 返回单个比特。该分布的均值位于 $[-\lambda,\lambda]$ 内，且其 $k$ 阶中心矩至多为 $\sigma^k$，其中 $k>1$ 固定。Lau 和 Scarlett 提出的阶最优两阶段协议利用第一批响应来选择第二批查询，这引出了一个问题：这一轮交互是否是必要的。我们对此给出否定回答：对于每个 $k>1$，存在一个非自适应协议能够达到自适应1比特极小极大速率（同时期的其他工作通过不同策略也得到了相同结论）。我们进一步确定了当每个一集合 $Q^{-1}(1)$ 被限制为至多 $s$ 个区间的并集时，非自适应1比特估计器的极小极大样本复杂度。相对于无限制的非自适应1比特查询，这一约束增加了一项阶为 $(\lambda\sigma/(s\varepsilon^2))\log(1/\delta)$ 的项，从而在 $k$ 相关常数因子内给出了样本复杂度与区间复杂度之间的完整权衡。作为推论，我们按阶确定了保持无限制1比特极小极大样本速率所需的最小区间预算。
英文摘要
We study distributed one-dimensional mean estimation under a 1-bit communication constraint. Each agent observes one sample, drawn independently from an unknown distribution, and returns a single bit in response to a query $Q: \mathbb{R}\to\{0,1\}$ chosen by a central learner. The distribution has mean in $[-λ,λ]$ and $k$-th central moment at most $σ^k$, for a fixed $k>1$. The order-optimal two-stage protocol of Lau and Scarlett uses responses from the first batch to choose the second-batch queries, motivating the question of whether this single round of interaction is necessary. We answer this negatively: for every $k>1$, a non-adaptive protocol attains the adaptive 1-bit minimax rate (and concurrent works reached the same conclusion via different strategies). We further determine the minimax sample complexity among non-adaptive 1-bit estimators when every one-set $Q^{-1}(1)$ is restricted to a union of at most $s$ intervals. Relative to unrestricted non-adaptive 1-bit querying, this constraint adds a term of order $(λσ/(s\varepsilon^2))\log(1/δ)$, giving the full tradeoff between sample complexity and interval complexity to within $k$-dependent constant factors. As a corollary, we identify, order-wise, the minimum interval budget needed to retain the unrestricted 1-bit minimax sample rate.
