Infinite population models are important tools for studying population dynamics of evolutionary algorithms. They describe how the distributions of populations change between consecutive generations. In general, infinite population models are derived from Markov chains by exploiting symmetries between individuals in the population and analyzing the limit as the population size goes to infinity. In this article, we study the theoretical foundations of infinite population models of evolutionary algorithms on continuous optimization problems. First, we show that the convergence proofs in a widely cited study were in fact problematic and incomplete. We further show that the modeling assumption of exchangeability of individuals cannot yield the transition equation. Then, in order to analyze infinite population models, we build an analytical framework based on convergence in distribution of random elements which take values in the metric space of infinite sequences. The framework is concise and mathematically rigorous. It also provides an infrastructure for studying the convergence of the stacking of operators and of iterating the algorithm which previous studies failed to address. Finally, we use the framework to prove the convergence of infinite population models for the mutation operator and the k-ary recombination operator. We show that these operators can provide accurate predictions for real population dynamics as the population size goes to infinity, provided that the initial population is identically and independently distributed.

中文翻译：

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对连续优化问题的进化算法的无限种群模型的重新审视

无限种群模型是研究进化算法种群动态的重要工具。他们描述了人口分布在连续几代之间如何变化。通常，无限种群模型是通过利用种群中个体之间的对称性并分析种群规模趋于无穷大时的极限而从马尔可夫链推导出来的。在本文中，我们研究了针对连续优化问题的进化算法的无限种群模型的理论基础。首先，我们表明一项广泛引用的研究中的收敛证明实际上是有问题且不完整的。我们进一步表明，个人可交换性的建模假设不能产生转移方程。然后，为了分析无限人口模型，我们构建了一个基于随机元素分布收敛的分析框架，这些元素在无限序列的度量空间中取值。该框架简洁且数学严谨。它还为研究运算符堆叠的收敛性和迭代先前研究未能解决的算法提供了基础设施。最后，我们使用该框架证明了变异算子和k-ary重组算子的无限种群模型的收敛性。我们表明，如果初始种群相同且独立分布，这些算子可以为实际种群动态提供准确的预测，因为种群规模趋于无穷大。该框架简洁且数学严谨。它还为研究运算符堆叠的收敛性和迭代先前研究未能解决的算法提供了基础设施。最后，我们使用该框架证明了变异算子和k-ary重组算子的无限种群模型的收敛性。我们表明，如果初始种群相同且独立分布，这些算子可以为实际种群动态提供准确的预测，因为种群规模趋于无穷大。该框架简洁且数学严谨。它还为研究运算符堆叠的收敛性和迭代先前研究未能解决的算法提供了基础设施。最后，我们使用该框架证明了变异算子和k-ary重组算子的无限种群模型的收敛性。我们表明，如果初始种群相同且独立分布，这些算子可以为实际种群动态提供准确的预测，因为种群规模趋于无穷大。我们使用该框架来证明变异算子和k元重组算子的无限种群模型的收敛性。我们表明，如果初始种群相同且独立分布，这些算子可以为实际种群动态提供准确的预测，因为种群规模趋于无穷大。我们使用该框架来证明变异算子和k元重组算子的无限种群模型的收敛性。我们表明，如果初始种群相同且独立分布，这些算子可以为实际种群动态提供准确的预测，因为种群规模趋于无穷大。