We introduce an acceleration for covariance matrix adaptation evolution strategies (CMA-ES) by means of adaptive diagonal decoding (dd-CMA). This diagonal acceleration endows the default CMA-ES with the advantages of separable CMA-ES without inheriting its drawbacks. Technically, we introduce a diagonal matrix D that expresses coordinate-wise variances of the sampling distribution in DCD form. The diagonal matrix can learn a rescaling of the problem in the coordinates within a linear number of function evaluations. Diagonal decoding can also exploit separability of the problem, but, crucially, does not compromise the performance on nonseparable problems. The latter is accomplished by modulating the learning rate for the diagonal matrix based on the condition number of the underlying correlation matrix. dd-CMA-ES not only combines the advantages of default and separable CMA-ES, but may achieve overadditive speedup: it improves the performance, and even the scaling, of the better of default and separable CMA-ES on classes of nonseparable test functions that reflect, arguably, a landscape feature commonly observed in practice. The article makes two further secondary contributions: we introduce two different approaches to guarantee positive definiteness of the covariance matrix with active CMA, which is valuable in particular with large population size; we revise the default parameter setting in CMA-ES, proposing accelerated settings in particular for large dimension. All our contributions can be viewed as independent improvements of CMA-ES, yet they are also complementary and can be seamlessly combined. In numerical experiments with dd-CMA-ES up to dimension 5120, we observe remarkable improvements over the original covariance matrix adaptation on functions with coordinate-wise ill-conditioning. The improvement is observed also for large population sizes up to about dimension squared.

中文翻译：

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协方差矩阵适应进化策略的对角线加速

我们通过自适应对角线解码 (dd-CMA) 引入协方差矩阵自适应演化策略 (CMA-ES) 的加速。这种对角线加速度赋予默认 CMA-ES 可分离 CMA-ES 的优点，而不会继承其缺点。从技术上讲，我们引入了一个对角矩阵 D，它以 DCD 形式表示采样分布的坐标方差。对角矩阵可以在线性数量的函数评估内学习坐标中问题的重新缩放。对角解码也可以利用问题的可分性，但至关重要的是，不会影响不可分问题的性能。后者是通过基于底层相关矩阵的条件数调制对角矩阵的学习率来实现的。dd-CMA-ES 不仅结合了 default 和 separable CMA-ES 的优点，而且可以实现过加性加速：它提高了 default 和 separable CMA-ES 在不可分测试函数类上的性能，甚至扩展性可以说，这反映了实践中普遍观察到的景观特征。这篇文章进一步做出了两个次要贡献：我们介绍了两种不同的方法来保证具有活动 CMA 的协方差矩阵的正定性，这在人口规模较大的情况下尤其有价值；我们修改了 CMA-ES 中的默认参数设置，特别针对大尺寸提出了加速设置。我们所有的贡献都可以看作是 CMA-ES 的独立改进，但它们也是互补的，可以无缝结合。在 dd-CMA-ES 高达 5120 维的数值实验中，我们观察到比原始协方差矩阵对具有坐标病态的函数的显着改进。对于高达大约平方维数的大群体，也观察到了改进。