This study targets the mixed-integer black-box optimization (MI-BBO) problem where continuous and integer variables should be optimized simultaneously. The CMA-ES, our focus in this study, is a population-based stochastic search method that samples solution candidates from a multivariate Gaussian distribution (MGD), which shows excellent performance in continuous BBO. The parameters of MGD, mean and (co)variance, are updated based on the evaluation value of candidate solutions in the CMA-ES. If the CMA-ES is applied to the MI-BBO with straightforward discretization, however, the variance corresponding to the integer variables becomes much smaller than the granularity of the discretization before reaching the optimal solution, which leads to the stagnation of the optimization. In particular, when binary variables are included in the problem, this stagnation more likely occurs because the granularity of the discretization becomes wider, and the existing modification to the CMA-ES does not address this stagnation. To overcome these limitations, we propose a simple extension of the CMA-ES based on lower-bounding the marginal probabilities associated with the generation of integer variables in the MGD. The numerical experiments on the MI-BBO benchmark problems demonstrate the efficiency and robustness of the proposed method. Furthermore, in order to demonstrate the generality of the idea of the proposed method, in addition to the single-objective optimization case, we incorporate it into multi-objective CMA-ES and verify its performance on bi-objective mixed-integer benchmark problems.

近年来，通过分布式数据的隐私保存的因果推断技术的开发引起了人们的关注。为了解决这个问题，我们提出了基于数据协作（DC-QE）的准实验，该实验可以从具有隐私保护的分布式数据中获得因果推断。我们的方法通过仅共享降低维度的中间表示来保留私人数据的隐私，这些中间表示由各方单独构建。此外，我们的方法可以减少随机错误和偏见，而现有方法只能减少治疗效果估计中的随机错误。通过对人工和现实世界数据的数值实验，我们确认我们的方法可以比单个分析得出更好的估计结果。随着我们方法的传播，可以将中间表示形式作为开放数据发布，以帮助研究人员找到因果关系并积累为知识库。