Non-Adaptive 1-Bit Mean Estimation: Minimax Rates and the Sample-Interval Tradeoff

• URL: http://arxiv.org/abs/2609.08564v1
• Date: Tue, 08 Sep 2026 10:52:52 GMT
• Title: Non-Adaptive 1-Bit Mean Estimation: Minimax Rates and the Sample-Interval Tradeoff
• Abstract summary: We study one-dimensional mean estimation under a 1-bit communication constraint.<n>We find that for every $k>1$, a non-adaptive protocol attains the adaptive 1-bit minimax rate.
• Score: 32.65125292684608
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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