Stationary Density of Stochastic Search Processes

The optimization of a cost function which has a number of local minima is a relevant subject in many important fields. For instance, the determination of the weights of learning machines depends in general on the solution of global optimization tasks (Haykin, 1999). A feature shared by almost all of the most common deterministic and stochastic algorithms for continuous non – linear optimization is that their performance is strongly affected by their starting conditions. Depending on the algorithm, the correct selection of an initial point or set of points have direct consequences on the efficiency, or even on the possibility to find the global minima. Of course, adequate selection of seeds implies prior knowledge on the structure of the optimization task. In the absence of prior information, a natural choice is to draw seeds from a uniform density defined over the search space. Knowledge on the problem can be gained through the exploration of this space. In this contribution is presented a method to estimate probability densities that describe the asymptotic behavior of general stochastic search processes over continuously differentiable cost functions. The relevance of such densities is that they give a description of the residence times over the different regions of the search space, after an infinitely long exploration. The preferred regions are those which minimize the cost globally, which is reflected in the asymptotic densities. In first instance, the resulting densities can be used to draw populations of points that are consistent with the global properties of the associated optimization tasks. BACKGROUND