Comprehensive characterization of the behaviors of estimation of distribution algorithms

Estimation of distribution algorithms (EDAs) are a successful example of how to use machine learning techniques for designing robust and efficient heuristic search algorithms. Understanding the relationship between EDAs and the space of optimization problems is a fundamental issue for the successful application of this type of algorithms. A step forward in this matter is to create a taxonomy of optimization problems according to the different behaviors that an EDA can exhibit. This paper substantially extends previous work in the proposal of a taxonomy of problems for univariate EDAs, mainly by generalizing those results to EDAs that are able to deal with multivariate dependences among the variables of the problem. Through the definition of an equivalence relation between functions, it is possible to partition the space of problems into equivalence classes in which the algorithm has the same behavior. We provide a sufficient and necessary condition to determine the equivalence between functions. This condition is based on a set of matrices which provides a novel encoding of the relationship between the function and the probabilistic model used by the algorithm. The description of the equivalent functions belonging to a class is studied in depth for EDAs whose probabilistic model is given by a chordal Markov network. Assuming this class of factorization, we unveil the intrinsic connection between the behaviors of EDAs and neighborhood systems defined over the search space. In addition, we carry out numerical simulations that effectively reveal the different behaviors of EDAs for the injective functions defined over the search space {0,1}3{0,1}3. Finally, we provide a novel approach to extend the analysis of equivalence classes to non-injective functions.