Текст наукової роботи на тему «An analysis of genetic algorithms using evolutionary dynamics»
?Матеріали Міжнародної конференції
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YJ \ K 658.512
AN ANALYSIS OF GENETIC ALGORITHMS USING EVOLUTIONARY
Abstract: A formalism is developed for studying Genetic Algorithms by considering the evolution of the distribution of fitness in the population. The effects of selection on the population are problem independent The formalism predicts the optimal amount of selection. The theory is found to be in good agreement with simulations. Metastability is a commonly observed phenomenon in many population-based dynamical systems. In such systems, including evolutionary search algorithms, models of biological evolution, and ecological and sociological systems, the state of a population is often described as the distribution of certain features of interest over the population. A commonly observed qualitative behavior is that the distribution of these features alternates between periods of stasis and sudden change. Extended periods of time in which the system seems to have stabilized on some feature distribution are interrupted by brief periods of sudden change. In natural systems and in many models, epochal behavior is undoubtedly the result of a complicated and poorly understood web of mechanisms. In this paper we identify the mechanism that underlies the occurrence of epochal behavior in a simplified genetic algorithm. In the mechanism's most general form, metastability is induced in an area of state space where the local "flow" of the dynamics is small compared to a scale set by the population's finite size. The dynamics becomes too weak to drive changes in the finite population. More specifically, we will see that the metastability can be associated with an "entropic barrier" that the finite population must pass in moving through the "slow" part of state space. Metastability due to such entropic barriers is contrasted here with the more traditional explanation of metastability as being induced by "fitness barriers" In the latter the population stabilizes around a
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local fitness optimum in sequence space and must pass through a "valley" of lower fitness to find a higher-fitness optimum. We believe that the generality and simplicity of the mechanism for metastability presented in this paper makes it likely to play a role in the occurrence of epochal dynamics in the more general and complicated cases outlined in the previous paragraph.
Genetic algorithms (GAs) are a class of stochastic search techniques, loosely based on ideas from biological evolution, that have been used successfully for a great variety of different problems However, the mechanisms that control the dynamics of a GA on a given problem are not well understood GAs are nonlinear population-based dynamical systems. The complicated dynamics exhibited by such systems has been appreciated in the field of mathematical population genetics for decades. These complications make an empirical approach to the question of when and how to use evolutionary search problematic. Our approach to understanding GAs is to examine the change in the distribution of energies in the population, pt (E), at each generation /. For a finite population the energy distribution is a sum of delta functions which will depend on the particular choice of the couplings and on the randomly chosen initial population. We will therefore consider the statistical Properties of the distribution, the mean, variance and higher cumulants which are simply related to the moments. The bulk of the remainder is devoted to the development of this theoretical model. We have termed our type of analysis "statistical dynamics", since it combines a dynamical systems approach, on the one hand, with a statistical physics and stochastic process approach, on the other. The infinite Population behavior is treated as the dynamics of a detentiims-tic nonlinear dynamical system In constructing this dynamical system we have to choose suitable "mesoscopic state variables (in this case, fitness distributions) that capture the complicated microscopic state of the system in terms of a * uch smaller number of variables. Moreover, we require that the desenpt.on of the system and its Savior in terms of these variables should be closed in the limit of infinite populations. That is for infinite populations the dynamics of fitness distributions must be fully specfied by the fitness distributions themselves and should not depend on the exact underlying (microscopic disfribut.on of strings. This condensation of the microscopic states using a few order parameters that describe 'he dynamics in the limit of infinite system size, is a well-known procedure from statistical physics. With this setting established, we augment the solution of the nonlinear dynamical system with a statistical treatment of the finite population behavior. In doing so we make use of simple stochastic differential equations, such as the Fokker-Planck equation These three features-describing the astern in terns of a mall set of statistical order parameters denving and solving the deterministic nonlinear dynamical systems equations in the infinite population l .mit, and hen augmenting this solution With simple stochastic differential equations to capture the finite population dynamics -is uie essence of our statistical dynamics approach. The approach adopted here gives both a qualitative and quantitative understanding of how GAs work. This allows some strong predtct.ons to be made about the optimal parameters and representations that should be used.
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Алгоритму розбивки ГРАФА НА ОСНОВІ ГЕНЕТИЧНОГО
в даній роботі пропонуються нові і модифіковані ГА для розбиття графа на подграфа [1 ^ 1 вони відрізняються застосуванням нетрадиційних архітектур генетичного Поіс «а, а також генетичних операторів, орієнтованих на використанні знань про
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