Towards the geometry of estimation of distribution algorithms based on the exponential family

In this paper we present a geometrical framework for the analysis of Estimation of Distribution Algorithms (EDAs) based on the exponential family. From a theoretical point of view, an EDA can be modeled as a sequence of densities in a statistical model that converges towards distributions with reduced support. Under this framework, at each iteration the empirical mean of the fitness function decreases in probability, until convergence of the population. This is the context of stochastic relaxation, i.e., the idea of looking for the minima of a function by minimizing its expected value over a set of probability densities. Our main interest is in the study of the gradient of the expected value of the function to be minimized, and in particular on how its landscape changes according to the fitness function and the statistical model used in the relaxation. After introducing some properties of the exponential family, such as the description of its topological closure and of its tangent space, we provide a characterization of the stationary points of the relaxed problem, together with a study of the minimizing sequences with reduced support. The analysis developed in the paper aims to provide a theoretical understanding of the behavior of EDAs, and in particular their ability to converge to the global minimum of the fitness function. The theoretical results of this paper, beside providing a formal framework for the analysis of EDAs, lead to the definition of a new class algorithms for binary functions optimization based on Stochastic Natural Gradient Descent (SNGD), where the estimation of the parameters of the distribution is replaced by the direct update of the model parameters by estimating the natural gradient of the expected value of the fitness function.

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