Resolution Limits for the Noisy Non-Adaptive 20 Questions Problem

• Published in: IEEE transactions on information theory Vol. 67; no. 4; pp. 2055 - 2073
• Main Authors: Zhou, Lin, Hero, Alfred O.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.04.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 20 questions; adaptive; Asymptotic methods; Asymptotic properties; Channel coding; Constraints; Entropy; finite blocklength analysis; Loss measurement; multidimensional target; Noise measurement; non-adaptive; Probability distribution; probably approximately correct learning; Queries; Questions; resolution; second-order asymptotics; sorted posterior matching; Transforms; Upper bound
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2021.3049796

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.