Given the ubiquity of non-separable optimization problems in real worlds, in this paper we analyze and extend the large-scale version of the well-known cooperative coevolution (CC), a divide-and-conquer black-box optimization framework, on non-separable functions. First, we reveal empirical reasons of when decomposition-based methods are preferred or not in practice on some non-separable large-scale problems, which have not been clearly pointed out in many previous CC papers. Then, we formalize CC to a continuous-game model via simplification, but without losing its essential property. Different from previous evolutionary game theory for CC, our new model provides a much simpler but useful viewpoint to analyze its convergence, since only the pure Nash equilibrium concept is needed and more general fitness landscapes can be explicitly considered. Based on convergence analyses, we propose a hierarchical decomposition strategy for better generalization, as for any decomposition, there is a risk of getting trapped into a suboptimal Nash equilibrium. Finally, we use powerful distributed computing to accelerate it under the recent multi-level learning framework, which combines the fine-tuning ability from decomposition with the invariance property of CMA-ES. Experiments on a set of high-dimensional test functions validate both its search performance and scalability (w.r.t. CPU cores) on a clustering computing platform with 400 CPU cores.

翻译：鉴于现实世界中不可分离优化问题的普遍性，本文分析和扩展了著名的合作协同进化（CC）这一分治黑箱优化框架在大规模不可分离函数上的应用。首先，我们揭示了以往许多CC论文中未明确指出的实践原因：在解决某些不可分离大规模问题时，何时应优先选用或避免分解类方法。随后，通过简化但不失本质特性，我们将CC形式化为连续博弈模型。与以往的进化博弈论分析不同，新模型仅需纯纳什均衡概念且可显式考虑更一般的适应度景观，为分析收敛性提供了更简洁但有效的视角。基于收敛性分析，我们提出分层分解策略以增强泛化能力——任何分解都存在陷入次优纳什均衡的风险。最后，利用强大的分布式计算，在最新的多层学习框架下加速该策略，该框架融合了分解的微调能力与CMA-ES的不变性特性。在包含400个CPU核心的集群计算平台上，对一组高维测试函数的实验验证了其搜索性能和可扩展性（关于CPU核心数）。