平均场律下各向异性基于共识优化的收敛性
Convergence of Anisotropic Consensus-Based Optimization in Mean-Field Law
AI总结：
本文研究了各向异性基于共识的优化（CBO）方法，证明了其在极小初始化假设下能以与维度无关的速率全局收敛，并揭示了智能体趋于无穷时CBO对优化问题的凸化机制。
摘要：
本文研究各向异性基于共识的优化（CBO），这是一种多智能体元启发式无导数优化方法，能够在高维空间中全局最小化非凸和非光滑函数。CBO基于随机群体智能，并受共识动力学和观点形成的启发。与粒子群优化等其他元启发式算法相比，CBO本质上更简单，因此更适合理论分析。通过采用一种最近建立的证明技术，我们证明了在方法初始化的极小假设下，对于一大类目标函数，各向异性CBO以与维度无关的速率全局收敛。此外，证明技术表明，随着智能体数量趋于无穷大，CBO对优化问题进行了凸化，从而深入洞察了导致该方法成功的内部CBO机制。为了从实践角度激发各向异性CBO，我们进一步在一个复杂的高维基准问题上测试了该方法，该问题在机器学习文献中得到了充分理解。
英文摘要：
In this paper we study anisotropic consensus-based optimization (CBO), a multi-agent metaheuristic derivative-free optimization method capable of globally minimizing nonconvex and nonsmooth functions in high dimensions. CBO is based on stochastic swarm intelligence, and inspired by consensus dynamics and opinion formation. Compared to other metaheuristic algorithms like particle swarm optimization, CBO is of a simpler nature and therefore more amenable to theoretical analysis. By adapting a recently established proof technique, we show that anisotropic CBO converges globally with a dimension-independent rate for a rich class of objective functions under minimal assumptions on the initialization of the method. Moreover, the proof technique reveals that CBO performs a convexification of the optimization problem as the number of agents goes to infinity, thus providing an insight into the internal CBO mechanisms responsible for the success of the method. To motivate anisotropic CBO from a practical perspective, we further test the method on a complicated high-dimensional benchmark problem, which is well understood in the machine learning literature.
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