Resolution Limits for the Noisy Non-Adaptive 20 Questions Problem
Abstract
We establish fundamental limits on estimation accuracy for the noisy 20 questions problem with measurement-dependent noise and introduce optimal non-adaptive procedures that achieve these limits. The minimal achievable resolution is defined as the absolute difference between the estimated and the true locations of a target over a unit cube, given a finite number of queries constrained by the excess-resolution probability. Inspired by the relationship between the 20 questions problem and the channel coding problem, we derive non-asymptotic bounds on the minimal achievable resolution to estimate the target location. Furthermore, applying the Berry--Esseen theorem to our non-asymptotic bounds, we obtain a second-order asymptotic approximation to the achievable resolution of optimal non-adaptive query procedures with a finite number of queries subject to the excess-resolution probability constraint. We specialize our second-order results to measurement-dependent versions of several channel models including the binary symmetric, the binary erasure and the binary Z- channels. As a complement, we establish a second-order asymptotic achievability bound for adaptive querying and use this to bound the benefit of adaptive querying.
Cite
@article{arxiv.2004.07231,
title = {Resolution Limits for the Noisy Non-Adaptive 20 Questions Problem},
author = {Lin Zhou and Alfred Hero},
journal= {arXiv preprint arXiv:2004.07231},
year = {2021}
}
Comments
To appear in IEEE Transactions on Information Theory, 2021. The conference version is v1, which appeared in ISIT 2020. In fact, this version is the latest version of arXiv:1909.12954, which includes all the revisions made during to the review progress and thus v2/v3 here should be named v4 of arXiv:1909.12954. This is the reason why there is a substantial text overlap