球面上基于粒子群的Optimization的各向异性扩散
Anisotropic Diffusion in Consensus-based Optimization on the Sphere
摘要：
本文利用多智能体无导数方法研究了约束在单位超球面上的可能非光滑非凸目标函数的全局极小化问题。该算法属于最近推出的基于神经网络的优化类。事实上，智能体在球体上移动，由朝向瞬时共识点的漂移驱动，瞬时共识点被计算为智能体位置的凸组合，根据拉普拉斯原理由费用函数加权，并且它表示对全局最小值的近似。各向异性随机向量场进一步扰动动力学，以利于勘探。本文的主要结果是在充分准备初值的条件下，证明了数值格式收敛于全局极小值。收敛性的证明结合了一个平均场极限的结果与一个新的渐近分析，和经典的收敛结果的数值方法的反问题。相对于以前的工作的主要创新是各向异性随机项的引入，这使我们能够确保从维度的算法的参数的独立性和规模的方法工作在非常高的维度。我们提出了几个数值实验，这表明，在本文中提出的算法是非常通用的，优于以前的配方与各向同性随机噪声。
英文摘要：
In this paper we are concerned with the global minimization of a possibly non-smooth and non-convex objective function constrained on the unit hypersphere by means of a multi-agent derivative-free method. The proposed algorithm falls into the class of the recently introduced Consensus-Based Optimization. In fact, agents move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of agent locations, weighted by the cost function according to Laplaces principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by an anisotropic random vector field to favor exploration. The main results of this paper are about the proof of convergence of the numerical scheme to global minimizers provided conditions of well-preparation of the initial datum. The proof of convergence combines a mean-field limit result with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. The main innovation with respect to previous work is the introduction of an anisotropic stochastic term, which allows us to ensure the independence of the parameters of the algorithm from the dimension and to scale the method to work in very high dimension. We present several numerical experiments, which show that the algorithm proposed in the present paper is extremely versatile and outperforms previous formulations with isotropic stochastic noise.
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