Truncation Selection and Gaussian EDA: Bounds for Sustainable Progress in High-Dimensional Spaces

Abstract

Abstract. In real-valued estimation-of-distribution algorithms, the Gaussian distribution is often used along with maximum likelihood (ML) estimation of its parameters. Such a process is highly prone to premature convergence. The simplest method for preventing premature convergence of Gaussian distribution is enlarging the maximum likelihood estimate of σ by a constant factor k each generation. Such a factor should be large enough to prevent convergence on slopes of the fitness function, but should not be too large to allow the algorithm converge in the neighborhood of the optimum. Previous work showed that for truncation selection such admissible k exists in 1D case. In this article it is shown experimentaly, that for the Gaussian EDA with truncation selection in high-dimensional spaces no admissible k exists!

Preventing Premature Convergence in a Simple EDA Via Global Step Size Setting

“…However, with increasing dimensionality, the value of k min grows faster than k max for all values of τ , and for dimensions greater then 5 there is no admissible k (which would ensure effective traversing of slopes and focusing to the optimum in the same time). This is in accordance with the results in [13] and [14]. The situation for C iso distribution is even better, see Fig.…”

“…In this article, an approach of [14] is used where the combined effect of the selection and variation is taken into account.…”

“…Although [13] theoretically deduced bounds for k in case of 1D Gaussian distribution, in [14] it was shown that the process sketched above does not work with increasing dimensionality, since the interval of admissible k diminishes and eventually vanishes. This is due to the fact that the variance after selection in the neighborhood of the valley (sphere function) increases with dimensionality, thus k max must be successively smaller and eventually gets lower than k min .…”

“…1, that suggests enlarging the population variance by a constant factor each generation, was studied in [12] where the minimal values of the 'amplification coefficient' were determined by a search in 1D case. In [13], the theoretical model of the algorithm behavior in 1D was used to derive the minimal and maximal admissible values for k. However, in [14] it was shown experimentaly that a constant multiplier does not ensure the desired properties of the algorithm when increasing the dimensionality of the search space.…”