Resolution Limits of Non-Adaptive 20 Questions Search for a Moving Target

TL;DR

This paper investigates the fundamental limits of non-adaptive 20 questions search strategies for tracking a moving target with unknown initial position and velocity, under noisy conditions, using information theory techniques.

Contribution

It derives tight bounds on the minimal resolution achievable with a finite number of queries in noisy, non-adaptive search for a moving target, extending to Gaussian noise.

Findings

Bounds are tight in first-order asymptotics for general cases.

Bounds are tight in second-order asymptotics for constant velocity targets.

The problem is related to channel coding and finite blocklength information theory.

Abstract

Using the 20 questions estimation framework with query-dependent noise, we study non-adaptive search strategies for a moving target over the unit cube with unknown initial location and velocities under a piecewise constant velocity model. In this search problem, there is an oracle who knows the instantaneous location of the target at any time. Our task is to query the oracle as few times as possible to accurately estimate the location of the target at any specified time. We first study the case where the oracle's answer to each query is corrupted by discrete noise and then generalize our results to the case of additive white Gaussian noise. In our formulation, the performance criterion is the resolution, which is defined as the maximal $L_\infty$ distance between the true locations and estimated locations. We characterize the minimal resolution of an optimal non-adaptive query procedure with a finite number of queries by deriving non-asymptotic and asymptotic bounds. Our bounds are tight in the first-order asymptotic sense when the number of queries satisfies a certain condition and our bounds are tight in the stronger second-order asymptotic sense when the target moves with a constant velocity. To prove our results, we relate the current problem to channel coding, borrow ideas from finite blocklength information theory and construct bounds on the number of possible quantized target trajectories.