This paper summarizes the research on population-based probabilistic search algorithms based on modeling promising solutions by estimating their probability distribution and using the constructed model to guide the further exploration of the search space. It settles the algorithms in the field of genetic and evolutionary computation where they have been originated. All methods are classified into a few classes according to the complexity of the class of models they use. Algorithms from each of these classes are briefly described and their strengths and weaknesses are discussed.

The paper describes the hierarchical Bayesian optimization algorithm which combines the Bayesian optimization algorithm, local structures in Bayesian networks, and a powerful niching technique. The proposed algorithm is able to solve hierarchical traps and other difficult problems very efficiently.

In this paper the optimization of additively decomposed discrete functions is investigated. For these functions genetic algorithms have exhibited a poor performance. First the schema theory of genetic algorithms is reformulated in probability theory terms. A schema denes the structure of a marginal distribution. Then the conceptual algorithm BEDA is introduced. BEDA uses a Boltzmann distribution to generate search points. From BEDA a new algorithm, FDA, is derived. FDA uses a factorization of the distribution. The factorization captures the structure of the given function. The factorization problem is closely connected to the theory of conditional independence graphs. For the test functions considered, the performance of FDA- in number of generations till convergence- is similar to that of a genetic algorithm for the OneMax function. This result is theoretically explained.

This paper proposes an algorithm that uses an estimation of the joint distribution of promising solutions in order to generate new candidate solutions. The algorithm is settled into the context of genetic and evolutionary computation and the algorithms based on the estimation of distributions. The proposed algorithm is called the Bayesian Optimization Algorithm (BOA). To estimate the distribution of promising solutions, the techniques for modeling multivariate data by Bayesian networks are used. TheBOA identifies, reproduces, and mixes building blocks up to a specified order. It is independent of the ordering of the variables in strings representing the solutions. Moreover, prior information about the problem can be incorporated into the algorithm, but it is not essential. First experiments were done with additively decomposable problems with both nonoverlapping as well as overlapping building blocks. The proposed algorithm is able to solve all but one of the tested problems in linear or close to linear time with respect to the problem size. Except for the maximal order of interactions to be covered, the algorithm does not use any prior knowledge about the problem. The BOA represents a step toward alleviating the problem of identifying and mixing building blocks correctly to obtain good solutions for problems with very limited domain information.

To solve hierarchical problems, one must be able to learn the linkage, represent partial solutions efficiently, and assure effective niching. We propose the hierarchical ...

Genetic and evolutionary algorithms have been increasingly applied to solve complex, large scale search problems with mixed success. Competent genetic algorithms have been proposed to solve hard problems quickly, reliably and accurately. They have rendered problems that were difficult to solve by the earlier GAs to be solvable, requiring only a subquadratic number of function evaluations. To facilitate solving large-scale complex problems, and to further enhance the performance of competent GAs, various efficiency-enhancement techniques have been developed. This study investigates one such class of efficiency-enhancement technique called evaluation relaxation. Evaluation-relaxation schemes replace a high-cost, low-error fitness function with a low-cost, high-error fitness function. The error in fitness functions comes in two flavors: Bias and variance. The presence of bias and variance in fitness functions is considered in isolation and strategies for increasing efficiency in both cases are developed. Specifically, approaches for choosing between two fitness functions with either differing variance or differing bias values have been developed. This thesis also investigates fitness inheritance as an evaluation-

FDA - the Factorized Distribution Algorithm - is an evolutionary algorithm which combines mutation and recombination by using a distribution instead. The distribution is estimated from a set of selected points. In general a discrete distribution defined for n binary variables has 2 n parameters. Therefore it is too expensive to compute. For additively decomposed discrete functions (ADFs) there exist algorithms which factor the distribution into conditional and marginal distributions. This factorization is used by FDA. The scaling of FDA is investigated theoretically and numerically. The scaling depends on the ADF structure and the specific assignment of function values. Difficult functions on a chain or a tree structure are solved in about O(n p n) operations. More standard genetic algorithms are not able to optimize these functions. FDA is not restricted to exact factorizations. It also works for approximate factorizations as is shown for a circle and a grid structure. By using results from Bayes networks, FDA is extended to LFDA. LFDA computes an approximate factorization using only the data, not the ADF structure. The scaling of LFDA is compared to the scaling of FDA. Keywords Genetic algorithms, Boltzmann distribution, simulated annealing, Bayes network, learning of Bayes networks, convergence, factorization of distributions. 1

The paper investigates the optimization of additively decomposable functions (ADF) by a new evolutionary algorithm called Factorized Distribution Algorithm (FDA). FDA is based on a factorization of the distribution to generate search points. First separable ADFs are considered. These are mapped to generalized linear functions with metavariables defined for multiple alleles. The mapping transforms FDA into an Univariate Marginal Frequency Algorithm (UMDA). For UMDA the exact equation for the response to selection is computed under the assumption of proportionate selection. For truncation selection an approximate equation for the time to convergence is used, derived from an analysis of the OneMax function. FDA is also numerically investigated for non separable functions. The time to convergence is very similar to separable ADFs. FDA outperforms the genetic algorithm with recombination of strings by far.

In this paper we perform a comparison among FSS-EBNA, a randomized, populationbased and evolutionary algorithm, and two genetic and other two sequential search approaches in the well known Feature Subset Selection (FSS) problem. In FSS-EBNA, the FSS problem, stated as a search problem, uses the EBNA (Estimation of Bayesian Network Algorithm) search engine, an algorithm within the EDA (Estimation of Distribution Algorithm) approach. The EDA paradigm is born from the roots of the GA community in order to explicitly discover the relationships among the features of the problem and not disrupt them by genetic recombination operators. The EDA paradigm avoids the use of recombination operators and it guarantees the evolution of the population of solutions and the discovery of these relationships by the factorization of the probability distribution of best individuals in each generation of the search. In EBNA, this factorization is carried out by a Bayesian network induced by a chea...

This paper discusses the use of various scoring metrics in the Bayesian optimization algorithm (BOA) which uses Bayesian networks to model promising solutions and generate the new ones. The use of decision graphs in Bayesian networks to improve the performance of the BOA is proposed. To favor simple models, a complexity measure is incorporated into the Bayesian-Dirichlet metric for Bayesian networks with decision graphs. The presented modi cations are compared on a number of interesting problems.