Bilevel optimization (BLO) offers a principled framework for hierarchical decision-making and has been widely applied in machine learning tasks such as hyperparameter optimization and meta-learning. While existing BLO methods are mostly developed in Euclidean spaces, many real-world problems involve structural constraints. In this paper, we propose a Riemannian bilevel optimization (RBLO) algorithm that incorporates a bilevel descent aggregation (BDA) scheme to jointly coordinate upper- and lower-level updates. Concretely, first we abstract the constraints in the BLO to a manifold structure and then transform the constrained BLO be a unconstrained RBLO problem. Second, to address limitations of existing RBLO methods, particularly the restrictive assumptions required for convergence, we reformulate the bilevel problem using smooth manifold mappings and provide a convergence analysis under the conditions of geodesic convexity and Lipschitz smoothness. Finally, we recall the multi-view hypergraph spectral clustering task, and evaluate the proposed approach on 3sources data sets. The numerical results validate the superior performance over Euclidean and manifold-based baselines.

翻译：双层优化（BLO）为分层决策提供了一个原则性框架，并已广泛应用于超参数优化和元学习等机器学习任务中。虽然现有的BLO方法大多在欧几里得空间中发展，但许多现实世界问题涉及结构约束。本文提出了一种黎曼双层优化（RBLO）算法，该算法结合了双层下降聚合（BDA）方案来联合协调上层和下层的更新。具体而言，首先我们将BLO中的约束抽象为流形结构，然后将约束BLO转化为无约束的RBLO问题。其次，针对现有RBLO方法的局限性，特别是收敛所需的严格假设，我们利用光滑流形映射重新表述了双层问题，并在测地凸性和Lipschitz光滑性条件下提供了收敛性分析。最后，我们回顾了多视图超图谱聚类任务，并在3sources数据集上评估了所提出的方法。数值结果验证了该方法相对于欧几里得和基于流形的基线方法的优越性能。