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

Abstract

Abstract. When a simple real-valued estimation of distribution algorithm (EDA) with Gaussian model and maximum likelihood estimation of parameters is used, it converges prematurely even on the slope of the fitness function. The simplest way of preventing premature convergence by multiplying the variance estimate by a constant factor k each generation is studied. Recent works have shown that when increasing the dimensionality of the search space, such an algorithm becomes very quickly unable to traverse the slope and focus to the optimum at the same time. In this paper it is shown that when isotropic distributions with Gaussian or Cauchy distributed norms are used, the simple constant setting of k is able to ensure a reasonable behaviour of the EDA on the slope and in the valley of the fitness function at the same time.

On the Parallel Speed-Up of Estimation of Multivariate Normal Algorithm and Evolution Strategies

“…UMDA [14], Compact Genetic Algorithm [10], Population-Based Incremental Learning [1], Relative Entropy [13], CrossEntropy [5] and Estimation of Multivariate Normal Algorithms (EMNA) [11] (our main inspiration), which combine (i) the current distribution (possibly), (ii) statistical properties of selected points, into a new distribution. We show in this paper that forgetting the old estimate and only using the new points is a good idea in the case of λ large; in particular, premature convergence as pointed out in [20,7,12,15] does not occur if λ >> 1 points are distributed on the search space with non-degenerated variance, and troubles around variance estimates for small sample size as in [6] are by definition not relevant for us. Its advantages are as follows for λ large: (i) it's very simple and parameter free; the reduced number of parameters is an advantage of mutative self adaptation in front of cumulative step-size adaptation, but we show here that yet fewer parameters (0!)…”

Multivariate Cauchy EDA Optimisation

“…This means that it is more prone to producing values that fall far from its mean, thus, giving Cauchy more chance of sampling further down its tail than the Gaussian. This gives Cauchy a higher chance of escaping premature convergence than the Gaussian [6], [10].…”

“…The premature convergence of classical Gaussian EDA attracted many efforts geared towards solving this problem [6], [10]. This paper presents the usage of Multivariate Cauchy distribution, an extension of [10] with a full matrix valued parameter that encodes dependencies between the search variable as an alternative search operator in EDA.…”

“…We utilize its capability of making long jumps so as to escape premature convergence, which is typical of Gaussian in order to enable EDA algorithms get to the global optimum. Although Cauchy has already been used as an alternative search distribution for EDA [4], [5], [6], [10], it was the univariate version of Cauchy that was utilized, discarding statistical dependences among the search variables. We compare the performance of Multivariate Gaussian EDA with Multivariate Cauchy EDA to establish whether and when the long jumps that Cauchy is able make will be advantageous.…”

“…In classical EDA, Gaussian distribution is used as the search operator to build a probabilistic model to fit the fittest individuals and create new individuals by sampling from the created model. It has been established that Gaussian EDA is prone to premature convergence [1], [2], [6] when its parameters are estimated using the maximum likelihood estimation (MLE) method. It converges too fast and does not get to the global optimum.…”