Resolution Limits for the Noisy Non-Adaptive 20 Questions Problem

TL;DR

This paper establishes fundamental limits and optimal non-adaptive procedures for the noisy 20 questions problem with measurement-dependent noise, providing non-asymptotic bounds and second-order asymptotic approximations for estimation accuracy.

Contribution

It derives non-asymptotic bounds and second-order asymptotic results for measurement-dependent noisy 20 questions, extending to multiple targets and comparing adaptive and non-adaptive strategies.

Findings

Optimal non-adaptive procedures achieve the fundamental limits.

Second-order asymptotic approximations provide precise resolution estimates.

Adaptive querying offers limited additional benefit over non-adaptive methods.

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. The theory is extended to simultaneous searching for multiple targets. As a complement, we establish a second-order asymptotic achievability bound for adaptive querying and use this to bound the benefit of adaptive querying.