Although some of the earliest Estimation of Distribution Algorithms (EDAs) utilized bivariate marginal distribution models, up to now, all discrete bivariate EDAs had one serious limitation: they were constrained to exploiting only a limited O(d) subset out of all possible \(O(d^{2})\) bivariate dependencies. As a first we present a family of discrete bivariate EDAs that can learn and exploit all \(O(d^{2})\) dependencies between variables, and yet have the same run-time complexity as their more limited counterparts. This family of algorithms, which we label DICE (DIscrete Correlated Estimation of distribution algorithms), is rigorously based on sound statistical principles, and particularly on a modelling technique from statistical physics: dichotomised multivariate Gaussian distributions. Initially (Lane et al. in European Conference on the Applications of Evolutionary Computation, Springer, 1999), DICE was trialled on a suite of combinatorial optimization problems over binary search spaces. Our proposed dichotomised Gaussian (DG) model in DICE significantly outperformed existing discrete bivariate EDAs; crucially, the performance gap increasingly widened as dimensionality of the problems increased. In this comprehensive treatment, we generalise DICE by successfully extending it to multary search spaces that also allow for categorical variables. Because correlation is not wholly meaningful for categorical variables, interactions between such variables cannot be fully modelled by correlation-based approaches such as in the original formulation of DICE. Therefore, here we extend our original DG model to deal with such situations. We test DICE on a challenging test suite of combinatorial optimization problems, which are defined mostly on multary search spaces. While the two versions of DICE outperform each other on different problem instances, they both outperform all the state-of-the-art bivariate EDAs on almost all of the problem instances. This further illustrates that these innovative DICE methods constitute a significant step change in the domain of discrete bivariate EDAs.

中文翻译：

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DICE：利用二进制和多重搜索空间中的所有双变量依赖性

尽管一些最早的分布估计算法（EDAs）使用双变量边际分布模型，但到目前为止，所有离散的双变量EDA都有一个严重的局限性：它们被限制为在所有可能的\（d）中仅利用有限的O（d）子集。O（d ^ {2}）\）二元依存关系。首先，我们介绍了一个离散的双变量EDA系列，它们可以学习和利用变量之间的所有 \（O（d ^ {2}）\）依赖关系，但其运行时复杂度与有限的对应变量相同。这个算法系列，我们将其标记为DICE（DI crete C或相关的E严格地基于合理的统计原理，特别是基于统计物理学的建模技术：二元多元高斯分布。最初（Lane等人在欧洲进化计算应用会议，Springer，1999年）中，DICE在二元搜索空间上针对一组组合优化问题进行了试验。我们在DICE中提出的二分法高斯（DG）模型明显优于现有的离散二元EDA；至关重要的是，随着问题范围的扩大，性能差距也越来越大。在这种全面的处理中，我们通过将DICE成功地扩展到多重搜索空间（也允许分类变量。由于相关性对于分类变量而言并非完全有意义，因此此类变量之间的交互无法通过基于相关性的方法（例如DICE的原始公式化）完全建模。因此，这里我们扩展了原始的DG模型来处理这种情况。我们在具有挑战性的组合优化问题测试套件上测试DICE，该组合套件主要在共同搜索空间中定义。虽然DICE的两个版本在不同的问题实例上彼此优于，但它们在几乎所有问题实例上都优于所有最新的双变量EDA。这进一步说明，这些创新的DICE方法构成了离散双变量EDA域中的重要步骤变化。