Gaussian mixture model of evolutionary algorithms

Abstract

This paper proposes a novel finite Gaussian mixture model to study the population dynamics of evolutionary algorithms on continuous optimization problems. While previous research taking on a dynamical system view has established the transition equation between the density functions of consecutive populations, the equation usually does not have closed-form solutions and can only be applied to very few optimization problems. In this paper, we address this issue by approximating both the population density function of each generation and the objective function by finite Gaussian mixtures. We show that by making such approximations the transition equation can be solved exactly and key statistics, such as the expected mean and the variance of fitness values of the population, can be calculated easily. We also prove that by choosing appropriate values of the parameters, the $L^1$-norm error between our model and the actual population density function can be made arbitrarily small, up until a predefined generation. We present experimental results to show that our model is useful in simulating and examining the dynamics of evolutionary algorithms.