Collaborative 20 Questions for Target Localization

Abstract

We consider the problem of 20 questions with noise for multiple players under the minimum entropy criterion [1] 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 [1], [2] 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.

Unequal Error Protection Querying Policies for the Noisy 20 Questions Problem

“…For adaptive sequential querying, greedy successive entropy minimization of a target variable is often proposed as a way to design a querying strategy for estimation of the target variable [8], [10]. Successive-entropy-minimization strategies select a binary query that maximally reduces the remaining uncertainty of the target variable at each round.…”

“…In this simulation, we checked five pairs of (k 1 , k 2 ) including (5, 4), (6,5), (6,6), (7, 7), and (8,8) at the fixed rate pair (R 1 , R 2 ) with the increasing number of queries. The number of Monte Carlo trials at each simulation point is equal to 3000.…”

“…Previous studies on optimal query design for the noisy 20 questions problem often sought to design queries that acquire observations minimizing the posterior uncertainty of the target variable, where uncertainty was quantified by the Shannon entropy [9], [3], [10], [8]. In these works, the utility of observation is quantified by the expected reduction of the entropy due to the observation.…”

“…This general setup of noisy 20 questions game and the optimal query design problem is of broad interest, arising in various areas, including active learning [3], [4], optimal sensing [5] and experimental design [6], [7], with diverse applications. For example, a target localization problem in a sensor network [8] can be modeled as a noisy 20 questions game where a player (agency) aims to locate a target by receiving query responses from sensors probing the region of interest.…”

“…For the non-adaptive case, on the other hand, the controller designs the sequence of queries ahead of time without access to future answers of the oracle. In general, the use of feedback in the adaptive design provides an information advantage, allowing a better error rate of convergence, but at the cost of higher query design complexity and the need for a feedback channel.Previous studies on optimal query design for the noisy 20 questions problem often sought to design queries that acquire observations minimizing the posterior uncertainty of the target variable, where uncertainty was quantified by the Shannon entropy [9], [3], [10], [8]. In these works, the utility of observation is quantified by the expected reduction of the entropy due to the observation.…”