Estimation of distribution algorithms and minimum relative entropy

In the field of optimization using probabilistic models of the search space, this thesis identifies and elaborates several advancements in which the principles of maximum entropy and minimum relative entropy from information theory are used to estimate a probability distribution. The probability distribution within the search space is represented by a graphical model (factorization, Bayesian network or junction tree). An estimation of distribution algorithm (EDA) is an evolutionary optimization algorithm which uses a graphical model to sample a population within the search space and then estimates a new graphical model from the selected individuals of the population. • So far, the Factorized Distribution Algorithm (FDA) builds a factorization or Bayesian network from a given additive structure of the objective function to be optimized using a greedy algorithm which only considers a subset of the variable dependencies. Important connections can be lost by this method. This thesis presents a heuristic subfunction merge algorithm which is able to consider all dependencies between the variables (as long as the marginal distributions of the model do not become too large). On a 2-D grid structure, this algorithm builds a pentavariate factorization which allows to solve the deceptive grid benchmark problem with a much smaller population size than the conventional factorization. Especially for small population sizes, calculating large marginal distributions from smaller ones using Maximum Entropy and iterative proportional fitting leads to a further improvement. • The second topic is the generalization of graphical models to loopy structures. Using the Bethe-Kikuchi approximation, the loopy graphical model (region graph) can learn the Boltzmann distribution of an objective function by a generalized belief propagation algorithm (GBP). It minimizes the free energy, a notion adopted from statistical physics which is equivalent to the relative entropy to the Boltzmann distribution. Previous attempts to combine the Kikuchi approximation with EDA have relied on an expensive Gibbs sampling procedure for generating a population from this loopy probabilistic model. In this thesis a combination with a factorization is presented which allows more efficient sampling. The free energy is generalized to incorporate the inverse temperature β. The factorization building algorithm mentioned above can be employed here, too.