Mean Estimation from One-Bit Measurements

TL;DR

This paper investigates the problem of estimating the mean of a symmetric log-concave distribution using only one-bit measurements per sample, analyzing different settings and their efficiencies.

Contribution

It introduces a comprehensive analysis of mean estimation with one-bit data across centralized, adaptive, and distributed settings, highlighting efficiency limits and the role of adaptivity.

Findings

No asymptotic penalty in centralized setting for quantization

Median-based estimator achieves optimal efficiency in adaptive setting

One round of adaptivity suffices for optimal mean-square error

Abstract

We consider the problem of estimating the mean of a symmetric log-concave distribution under the constraint that only a single bit per sample from this distribution is available to the estimator. We study the mean squared error as a function of the sample size (and hence the number of bits). We consider three settings: first, a centralized setting, where an encoder may release $n$ bits given a sample of size $n$, and for which there is no asymptotic penalty for quantization; second, an adaptive setting in which each bit is a function of the current observation and previously recorded bits, where we show that the optimal relative efficiency compared to the sample mean is precisely the efficiency of the median; lastly, we show that in a distributed setting where each bit is only a function of a local sample, no estimator can achieve optimal efficiency uniformly over the parameter space. We additionally complement our results in the adaptive setting by showing that \emph{one} round of adaptivity is sufficient to achieve optimal mean-square error.