Resolution Limits of Non-Adaptive 20 Questions Search for Multiple Targets

TL;DR

This paper establishes fundamental limits on the resolution achievable by non-adaptive 20 questions search strategies for multiple targets in a multidimensional space, using information theory tools to derive non-asymptotic bounds.

Contribution

It extends previous first-order asymptotic results to non-asymptotic and second-order bounds for multi-target search, considering channels beyond the binary symmetric case.

Findings

Derived non-asymptotic bounds on resolution

Related search problem to multiple access and point-to-point channels

Extended analysis to channels beyond binary symmetric

Abstract

We study the problem of simultaneous search for multiple targets over a multidimensional unit cube and derive fundamental resolution limits of non-adaptive querying procedures using the 20 questions estimation framework. The performance criterion that we consider is the achievable resolution, which is defined as the maximal $L_\infty$ norm between the location vector and its estimated version where the maximization is over all target location vectors. The fundamental resolution limit is defined as the minimal achievable resolution of any non-adaptive query procedure, where each query has binary yes/no answers. We drive non-asymptotic and second-order asymptotic bounds on the minimal achievable resolution, using tools from finite blocklength information theory. Specifically, in the achievability part, we relate the 20 questions problem to data transmission over a multiple access channel, use the information spectrum method by Han and borrow results from finite blocklength analysis for random access channel coding. In the converse part, we relate the 20 questions problem to data transmission over a point-to-point channel and adapt finite blocklength converse results for channel coding. Our results extend the purely first-order asymptotic analyses of Kaspi \emph{et al.} (ISIT 2015) for the one-dimensional case: we consider channels beyond the binary symmetric channel and derive non-asymptotic and second-order asymptotic bounds on the performance of optimal non-adaptive query procedures.