In this paper, we study the problem of mean estimation under 1-bit communication constraints. We propose a novel adaptive mean estimator based solely on randomized threshold queries, where each 1-bit outcome indicates whether a given sample exceeds a sequentially chosen threshold. Our estimator is $(ε, δ)$-PAC for any distribution with a bounded mean $μ\in [-λ, λ]$ and a bounded $k$-th central moment $\mathbb{E}[|X-μ|^k] \le σ^k$ for any fixed $k > 1$. Moreover, our sample complexity is order-optimal in all such tail regimes, i.e., for every such $k$ value. For $k \neq 2$, our estimator's sample complexity matches the unquantized minimax lower bounds plus an unavoidable $O(\log(λ/σ))$ localization cost. For the finite-variance case ($k=2$), our estimator's sample complexity has an extra multiplicative $O(\log(σ/ε))$ penalty, and we establish a novel information-theoretic lower bound showing that this penalty is a fundamental limit of 1-bit quantization. We also establish a significant adaptivity gap: for both threshold queries and more general interval queries, the sample complexity of any non-adaptive estimator must scale linearly with the search space parameter $λ/σ$, rendering it vastly less sample efficient than our adaptive approach. Finally, we present algorithmic variants that (i) handle an unknown sampling budget, (ii) adapt to an unknown scale parameter $σ$ given (possibly loose) bounds, (iii) require only two stages of adaptivity to achieve order-optimal sample complexity at the expense of more general 1-bit queries, and (iv) leverage multiple local samples per 1-bit query to proportionally reduce communication costs.

翻译：本文研究1比特通信约束下的均值估计问题。我们提出一种基于随机阈值查询的新型自适应均值估计器，其中每个1比特输出指示给定样本是否超过依次选择的阈值。对于任意具有有界均值$μ\in [-λ, λ]$和有界$k$阶中心矩$\mathbb{E}[|X-μ|^k] \le σ^k$（固定$k>1$）的分布，我们的估计器满足$(ε, δ)$-PAC保证。此外，在所有此类尾分布条件下（即每个$k$值），样本复杂度均达到阶数最优。对于$k \neq 2$情形，估计器的样本复杂度匹配未量化极小化下界，并包含不可避免的$O(\log(λ/σ))$定位代价。对于有限方差情形（$k=2$），估计器的样本复杂度存在额外的乘性$O(\log(σ/ε))$惩罚项，我们通过建立新的信息论下界证明该惩罚是1比特量化的本质极限。我们还发现显著的适应性差距：对于阈值查询和更一般的区间查询，任何非自适应估计器的样本复杂度必须随搜索空间参数$λ/σ$线性增长，导致其样本效率远低于我们的自适应方法。最后，我们提出若干算法变体，包括：(i)处理未知采样预算的情形，(ii)给定（可能宽松的）边界下自适应未知尺度参数$σ$，(iii)仅需两阶段自适应性即可在牺牲更通用1比特查询的条件下实现阶数最优样本复杂度，(iv)利用每个1比特查询的多个局部样本按比例降低通信成本。