Achievable Resolution Limits for the Noisy Adaptive 20 Questions Problem

TL;DR

This paper investigates the limits of adaptive querying strategies for noisy 20 questions problems with measurement-dependent noise, deriving achievable resolution bounds and demonstrating improvements over existing algorithms through theoretical analysis and simulations.

Contribution

It introduces a new adaptive query procedure based on variable length feedback codes and shows its superior performance compared to state-of-the-art algorithms.

Findings

The proposed method achieves better resolution bounds at moderate to large excess-resolution probabilities.

Numerical simulations confirm the theoretical performance improvements.

The termination strategy significantly enhances asymptotic performance.

Abstract

We study the achievable performance of adaptive query procedures for the noisy 20 questions problem with measurement-dependent noise over a unit cube of finite dimension. The performance criterion that we consider is the minimal resolution, defined as the $L_\infty$ norm between the estimated and the true values of the random location vector of a target, given a finite number of queries constrained by an excess-resolution probability. Specifically, we derive the achievable resolution of an adaptive query procedure based on the variable length feedback code by Polyanskiy \emph{et al.} (TIT 2011). Furthermore, we verify our theoretical results with numerical simulations and compare the performance of our considered adaptive query procedure with that of certain state-of-the-art algorithms, such as the sorted posterior matching algorithm by Chiu and Javadi (ITW 2016). In particular, we demonstrate that the termination strategy adopted in our adaptive query procedure can significantly enhance the asymptotic performance of adaptive query procedures, especially at moderate to large excess-resolution probability constraints.