抽象黎曼流形上基于种群的随机优化框架
A Framework for Population-Based Stochastic Optimization on Abstract Riemannian Manifolds
AI总结：
该研究提出黎曼流形上的新型种群随机优化算法Extended RSDFO，结合信息几何与黎曼几何构建框架，证明其单调提升与有限步全局收敛性，在多流形多模态问题及Jacob's ladder实验中验证了方法优势。
摘要：
我们提出了扩展黎曼随机无导数优化（Extended RSDFO），这是一种新型的黎曼流形上基于种群的随机优化算法，解决了现有文献中流形优化的局部性问题及隐含假设。\n 我们首先研究了黎曼流形上统计模型的信息几何结构。该结构利用决策空间的统计几何与搜索空间的黎曼几何，构建了Extended RSDFO的几何框架。我们通过保定向微分同胚丛映射构造局部继承概率分布，随后将信息几何结构扩展到流形完全有界子集上的混合密度。前者将决策空间的信息几何与搜索空间流形上的局部点估计关联起来，后者则克服了黎曼流形上参数概率分布的局部性局限。\n 接着我们构建了Extended RSDFO，并从几何角度研究其结构与性质。我们证明，在连通紧黎曼流形上，Extended RSDFO的期望适应度单调提升，且能在有限步内实现全局最终收敛。\n 我们在多种流形上的多模态优化问题中，将Extended RSDFO与当前最先进的流形优化算法进行了对比。\n 特别地，我们在Jacob's ladder（雅各布天梯）上开展了一项新颖的合成实验，以说明流形优化的必要性与价值。Jacob's ladder是一种具有可数无限亏格的非紧流形，无法表示为多项式约束，也不存在欧几里得外围空间中的全局表示。因此，Jacob's ladder上的优化问题无法通过传统的欧氏空间（约束）优化方法求解。
英文摘要：
We present Extended Riemannian Stochastic Derivative-Free Optimization (Extended RSDFO), a novel population-based stochastic optimization algorithm on Riemannian manifolds that addresses the locality and implicit assumptions of manifold optimization in the literature. We begin by investigating the Information Geometrical structure of statistical model over Riemannian manifolds. This establishes a geometrical framework of Extended RSDFO using both the statistical geometry of the decision space and the Riemannian geometry of the search space. We construct locally inherited probability distribution via an orientation-preserving diffeomorphic bundle morphism, and then extend the information geometrical structure to mixture densities over totally bounded subsets of manifolds. The former relates the information geometry of the decision space and the local point estimations on the search space manifold. The latter overcomes the locality of parametric probability distributions on Riemannian manifolds. We then construct Extended RSDFO and study its structure and properties from a geometrical perspective. We show that Extended RSDFO's expected fitness improves monotonically and it's global eventual convergence in finitely many steps on connected compact Riemannian manifolds. Extended RSDFO is compared to state-of-the-art manifold optimization algorithms on multi-modal optimization problems over a variety of manifolds. In particular, we perform a novel synthetic experiment on Jacob's ladder to motivate and necessitate manifold optimization. Jacob's ladder is a non-compact manifold of countably infinite genus, which cannot be expressed as polynomial constraints and does not have a global representation in an ambient Euclidean space. Optimization problems on Jacob's ladder thus cannot be addressed by traditional (constraint) optimization methods on Euclidean spaces.
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