Order-Optimal Sequential 1-Bit Mean Estimation in General Tail Regimes

TL;DR

This paper introduces an order-optimal adaptive 1-bit mean estimator that achieves near-minimax sample complexity across various tail regimes, demonstrating fundamental limits of 1-bit quantization.

Contribution

It proposes a novel adaptive estimator based on randomized threshold queries that is order-optimal in all tail regimes and establishes fundamental lower bounds for 1-bit quantized mean estimation.

Findings

Estimator matches unquantized minimax lower bounds for k ≠ 2.

For k=2, a new lower bound shows unavoidable logarithmic penalty.

Adaptive approach significantly outperforms non-adaptive estimators in sample efficiency.

Abstract

In this paper, we study the problem of mean estimation under strict 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 $(\epsilon, \delta)$-PAC for any distribution with a bounded mean $\mu \in [-\lambda, \lambda]$ and a bounded $k$-th central moment $\mathbb{E}[|X-\mu|^k] \le \sigma^k$ for any fixed $k > 1$. Crucially, 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 $O(\log(\lambda/\sigma))$ localization cost. For the finite-variance case ($k=2$), our estimator's sample complexity has an extra multiplicative $O(\log(\sigma/\epsilon))$ 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 $\lambda/\sigma$, 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~$\sigma$ given (possibly loose) bounds, and (iii) require only two stages of adaptivity at the expense of more complicated general 1-bit queries.