Open Problem: Is Interaction Necessary for Order-Optimal 1-bit Mean Estimation?

2026-07-02

AI Summary

This study addresses the open question of whether interaction is necessary to achieve minimax optimal one-bit mean estimation over nonparametric classes with finite moments. Focusing on single-round communication protocols, the work formalizes this problem and compares the performance of non-adaptive and two-stage adaptive strategies through minimax analysis within a general framework for one-bit quantizers. The findings reveal that existing non-adaptive approaches—such as thresholding or interval queries—are substantially suboptimal, whereas a two-stage protocol with just one round of adaptation attains the optimal rate up to constants. However, it remains unresolved whether any non-adaptive quantizer can match this performance, thereby challenging prevailing assumptions about the necessity of adaptivity in achieving order-optimal estimation under one-bit constraints.

Abstract

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?

Problem

• 1-bit mean estimation
• interaction
• adaptive querying
• non-adaptive quantizers
• minimax rate

Innovation

• 1-bit mean estimation
• adaptive querying
• minimax rate
• nonparametric finite-moment classes
• non-adaptive quantizers