Measurement dependent noisy search: The Gaussian case

This paper considers the problem of searching for the unknown location of a target among a finite number of possible locations by probing multiple locations simultaneously. Outcome of each search measurement is corrupted by Gaussian noise whose intensity is proportional to the number of locations probed. We characterize a non-asymptotic lower bound on adaptivity gain; i.e. reduction in the expected number of measurements under an adaptive search strategies over the non-adaptive search strategies. Then we investigate the adaptivity gain in two complementary asymptotic regimes: one where the total search area is kept fixed but the location width is shrinking or the search resolution is increasing, and the other where each location width is fixed but the total search area is growing. Interestingly, adaptivity gain grows in distinctly different manner in these two regimes. In particular, adaptivity gains are significant in the later regime when the total search space grows; implying adaptivity is far more critical when either total search area or the noise intensity is large.