Resolution Limits for the Noisy Non-Adaptive 20 Questions Problem

• Published in: arXiv.org
• Main Authors: Zhou, Lin, Hero, Alfred
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 11.01.2021
• Subjects: Asymptotic properties; Computer Science - Information Theory; Constraints; Mathematics - Information Theory; Noise measurement; Queries; Questions; Random variables
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2004.07231

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.