This paper proposes a natural evolution strategy (NES) for mixed-integer black-box optimization (MI-BBO) that appears in real-world problems such as hyperparameter optimization of machine learning and materials design. This problem is difficult to optimize because plateaus where the values do not change appear when the integer variables are relaxed to the continuous ones. CMA-ES w. Margin that addresses the plateaus reportedly showed good performance on MI-BBO benchmark problems. However, it has been observed that the search performance of CMA-ES w. Margin deteriorates when continuous variables contribute more to the objective function value than integer ones. In order to address the problem of CMA-ES w. Margin, we propose Distance-weighted eXponential Natural Evolution Strategy taking account of Implicit Constraint and Integer (DX-NES-ICI). We compare the search performance of DX-NES-ICI with that of CMA-ES w. Margin through numerical experiments. As a result, DX-NES-ICI was up to 3.7 times better than CMA-ES w. Margin in terms of a rate of finding the optimal solutions on benchmark problems where continuous variables contribute more to the objective function value than integer ones. DX-NES-ICI also outperformed CMA-ES w. Margin on problems where CMA-ES w. Margin originally showed good performance.

翻译：本文提出了一种用于混合整数黑箱优化的自然进化策略，该策略可应用于机器学习超参数优化和材料设计等实际问题。由于整数变量被松弛为连续变量时会出现值不变的平台区域，此类问题难以优化。针对该平台问题的带边际CMA-ES在混合整数黑箱优化基准问题上展现出良好性能。然而研究表明，当连续变量对目标函数值的贡献大于整数变量时，带边际CMA-ES的搜索性能会下降。为解决带边际CMA-ES存在的问题，我们提出了考虑隐式约束与整数变量的距离加权指数自然进化策略。通过数值实验将DX-NES-ICI与带边际CMA-ES的搜索性能进行比较。结果表明，在连续变量对目标函数值的贡献大于整数变量的基准问题上，DX-NES-ICI的最优解发现率比带边际CMA-ES最高提升3.7倍。而在带边际CMA-ES原本表现优异的测试问题上，DX-NES-ICI同样展现出更优性能。