Structural Coherence of Problem and Algorithm: An Analysis for EDAs on all 2-bit and 3-bit Problems

Metaheuristics assume some kind of coherence between decision and objective spaces. Estimation of Distribution algorithms approach this by constructing an explicit probabilistic model of high fitness solutions, the structure of which is intended to reflect the structure of the problem. In this context, ``structure'' means the dependencies or interactions between problem variables in a probabilistic graphical model. There are many approaches to discovering these dependencies, and existing work has already shown that often these approaches discover ``unnecessary'' elements of structure - that is, elements which are not needed to correctly rank solutions. This work performs an exhaustive analysis of all 2 and 3 bit problems, grouped into classes based on mononotic invariance. It is shown in (Christie, 2014) that each class has a minimal Walsh structure that can be used to solve the problem. We compare the structure discovered by different structure learning approaches to the minimal Walsh structure for each class, with summaries of which interactions are (in)correctly identified. Our analysis reveals a large number of symmetries that may be used to simplify problem solving. We show that negative selection can result in improved coherence between discovered and necessary structure, and conclude with some directions for a general programme of study building on this work