AI 中文总结
研究基于各向异性共识的优化方法（CBO），该方法能利用高维目标函数可加分离结构减轻维度诅咒。证明其计算复杂度仅指数依赖内在维度\(\mathbf{d}\)，数值实验验证理论结果，突出相关因素对算法性能和复杂度的影响。
AI 中文摘要
基于各向异性共识的优化（CBO）是一种多智能体元启发式无导数优化方法，能可靠地找到非光滑和非凸目标函数的全局最小值，同时便于进行严格的理论分析。它能自动检测并利用高维目标函数的可加分离结构，减轻维度诅咒。本文证明各向异性CBO的计算复杂度仅指数依赖于目标函数的内在维度\(\mathbf{d}\)，而非环境维度\(D\gg\mathbf{d}\)。还表明计算复杂度仅取决于低维分量的可处理性条件。数值实验验证了理论结果，突出了内在维度\(\mathbf{d}\)、可分离性水平以及可分离分量中目标的复杂性和非凸性对各向异性CBO算法性能和计算复杂度的影响。
英文摘要
Anisotropic consensus-based optimization (CBO), a multi-agent metaheuristic derivative-free optimization method, which reliably finds global minima of nonsmooth and nonconvex objective functions while being amenable to rigorous theoretical analysis, automatically detects and exploits additively separable structures of high-dimensional objective functions. This enables the algorithm to mitigate the curse of dimensionality where the objective function decomposes additively into lower-dimensional components. In this paper, we show this property proving that the computational complexity of anisotropic CBO depends exponentially only on the intrinsic dimension $\mathbf{d}$ of the objective function, rather than the ambient dimension $D\gg\mathbf{d}$. Additionally, we demonstrate that the computational complexity depends only on tractability conditions of the lower-dimensional components rather than on the full energy landscape, allowing for a more refined description of the objective function and algorithmic complexity, as the objective function landscape is captured directly at the level of the individual components. Our results highlight the effectiveness of anisotropic CBO for additively separable objective functions provided sufficient alignment between the structure of the anisotropic noise and the separability structure of the objective. This motivates the design of an enhanced algorithm that learns during optimization how to effectively explore the loss landscape by aligning the noise with the structure of the objective function, which we leave for future research. Numerical experiments validate our theoretical results, accentuating the influence of the intrinsic dimensionality $\mathbf{d}$, the level of separability, and the complexity and non-convexity of the objective within the separable components on performance and computational complexity of the anisotropic CBO algorithm.
