Collaborative 20 Questions for Target Localization

TL;DR

This paper develops and compares sequential and joint questioning strategies for noisy 20 Questions problems involving multiple players, with applications to target localization and active machine learning, establishing their equivalence and analyzing convergence.

Contribution

It introduces a unified framework for multi-player noisy 20 Questions, characterizes optimal policies, and proves the equivalence of sequential and joint schemes in stochastic search.

Findings

Joint and sequential policies perform equally well on average.

Convergence rates for mean-square error are established.

Extension to unknown error probabilities is provided.

Abstract

We consider the problem of 20 questions with noise for multiple players under the minimum entropy criterion in the setting of stochastic search, with application to target localization. Each player yields a noisy response to a binary query governed by a certain error probability. First, we propose a sequential policy for constructing questions that queries each player in sequence and refines the posterior of the target location. Second, we consider a joint policy that asks all players questions in parallel at each time instant and characterize the structure of the optimal policy for constructing the sequence of questions. This generalizes the single player probabilistic bisection method for stochastic search problems. Third, we prove an equivalence between the two schemes showing that, despite the fact that the sequential scheme has access to a more refined filtration, the joint scheme performs just as well on average. Fourth, we establish convergence rates of the mean-square error (MSE) and derive error exponents. Lastly, we obtain an extension to the case of unknown error probabilities. This framework provides a mathematical model for incorporating a human in the loop for active machine learning systems.