Pure Random Search and Pure Adaptive Search

Abstract

A theoretical analysis of the performance of two stochastic methods is presented, pure random search (PRS) and pure adaptive search (PAS). These two random search algorithms can be viewed as extremes and are not intended to be practical algorithms, but rather are used to provide insight and bracket the performance of stochastic algorithms. Pure random search samples points from the domain independently, and the objective function has no impact on the technique of generating the next sample point. In contrast, pure adaptive search samples the next point from the subset of the domain with strictly superior objective function values. Pure random search is extreme in the sense that the iterates are completely independent and never use previous information to affect the search strategy, while pure adaptive search is extreme in an opposite sense because the iterates completely depend on each other, and the search strategy forces improvement by definition. These two extreme stochastic methods are analyzed and it is shown that, under certain conditions, the expected number of iterations of pure adaptive search is linear in dimension, while pure random search is exponential in dimension. The linearity result for pure adaptive search implies that adapting the search to sample improving points is very powerful.