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Life sciences · Preprint
arXiv · September 8, 2026
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This is a theoretical paper in distributed estimation and information theory that establishes minimax sample complexity rates for non-adaptive 1-bit mean estimation protocols. It proves that non-adaptive protocols can match adaptive ones (answering a protocol-design question) and characterises a tradeoff between sample complexity and interval constraint complexity. The work is mathematical in nature and does not address clinical, empirical, or practical problems.
Preprint.
Non-adaptive 1-bit protocols attain the adaptive minimax rate for all k>1 Interval constraint of at most s intervals adds sample complexity term of order (λσ/(sε²))log(1/δ) Identifies the minimum interval budget required to retain unrestricted 1-bit minimax sample rate
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This is a theoretical computer science paper establishing minimax rates and complexity bounds for a distributed estimation problem; it answers a mathematical question about protocol design rather than providing empirical evidence for clinical or practical application.
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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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