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

We ask whether interaction is necessary for order-optimal 1-bit mean estimation over nonparametric finite-moment classes. Adaptive threshold-query protocols achieve the order-optimal 1-bit minimax rate, and the same rate is attainable with general 1-bit queries using only one adaptive transition (i.e., two stages of querying). In the non-adaptive setting, threshold and interval queries are known to be highly suboptimal, but the case of arbitrary non-adaptive quantizers remains unresolved. Can such quantizers match the adaptive rate, yielding an optimal one-shot protocol? Or is the known two-stage estimator stage-optimal, with a single adaptive transition being necessary and sufficient?

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.

We characterize the fundamental limits of high-dimensional mean testing under arbitrary truncation, where samples are drawn from the conditional distribution $P(\cdot \mid S)$ for an unknown truncation set $S$ that may hide up to an $\varepsilon$-fraction of the probability mass. For distributions with $p$-th directional moments of magnitude at most $ν_{P,p}$, truncation induces a bias of order $O(ν_{P,p}\varepsilon^{1-1/p})$. This bias creates a sharp information-theoretic detectability floor: when the signal $α$ falls below this threshold, the null and alternative hypotheses are indistinguishable even with infinite data. Above this floor, we prove that a simple second-order test achieving near-optimal sample complexity $n = O\!\left(\frac{\|Σ_P\|}{(α-4ν_{P,p}\varepsilon^{1-1/p})^2}\sqrt{d}\right)$. We further identify a structural escape from this finite-moment bias barrier. Under a directional median regularity assumption, truncation bias improves to linear order $O(\varepsilon)$. This reveals an intermediate regime in which estimation requires $Θ(d)$ samples for uniform recovery, while testing recovers the classical $Θ(\sqrt d)$ rate once truncation bias is eliminated. Together, our results provide a unified framework for mean testing under truncation, connecting finite-moment, sub-Gaussian, and median-regular structural regimes.