Random Walk-Based Solution to Triple Level Stochastic Point Location Problem

This paper considers the stochastic point location (SPL) problem as a learning mechanism trying to locate a point on a real line via interacting with a random environment. Compared to the stochastic environment in the literatures that confines the learning mechanism to moving in two directions, i.e., left or right, this paper introduces a general triple level stochastic environment which not only tells the learning mechanism to go left or right, but also informs it to stay unmoved. It is easy to understand, as we will prove in this paper, that the environment reported in the previous literatures is just a special case of the triple level environment. And a new learning algorithm, named as random walk-based triple level learning algorithm, is proposed to locate an unknown point under this new type of environment. In order to examine the performance of this algorithm, we divided the triple level SPL problems into four distinguished scenarios by the properties of the unknown point and the stochastic environment, and proved that even under the triple level nonstationary environment and the convergence condition having not being satisfied for some time, which are rarely considered in existing SPL problems, the proposed learning algorithm is still working properly whenever the unknown point is static or evolving with time. Extensive experiments validate our theoretical analyses and demonstrate that the proposed learning algorithms are quite effective and efficient.