Noisy searching: simple, fast and correct
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This work revisits the multiplicative weights update technique (MWU) which has a variety of applications, especially in learning and searching algorithms. In particular, the Bayesian update method is a well known version of MWU that is particularly applicable for the problem of searching in a given domain. An ideal scenario for that method is when the input distribution is known a priori and each single update maximizes the information gain. In this work we consider two search domains - linear orders (sorted arrays) and graphs, where the aim of the search is to locate an unknown target by performing as few queries as possible. Searching such domains is well understood when each query provides a correct answer and the input target distribution is uniform. Hence, we consider two generalizations: the noisy search both with arbitrary and adversarial (i.e., unknown) target distributions. We obtain several results providing full characterization of the query complexities in the three settings: adversarial Monte Carlo, adversarial Las Vegas and distributional Las Vegas. Our algorithms either improve, simplify or patch earlier ambiguities in the literature - see the works of Emamjomeh-Zadeh et al. [STOC 2016], Dereniowski et. al. [SOSA@SODA 2019] and Ben-Or and Hassidim [FOCS 2008]. In particular, all algorithms give strategies that provide the optimal number of queries up to lower-order terms. Our technical contribution lies in providing generic search techniques that are able to deal with the fact that, in general, queries guarantee only suboptimal information gain.
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