Riemannian Bilevel Optimization with Gradient Aggregation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908599126589440 |
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| author | Chen, Zhuo Xu, Xinjian Ying, Shihui Zeng, Tieyong |
| author_facet | Chen, Zhuo Xu, Xinjian Ying, Shihui Zeng, Tieyong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15305 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Riemannian Bilevel Optimization with Gradient Aggregation Chen, Zhuo Xu, Xinjian Ying, Shihui Zeng, Tieyong Optimization and Control Numerical Analysis 49Q12, 65K10 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. |
| title | Riemannian Bilevel Optimization with Gradient Aggregation |
| topic | Optimization and Control Numerical Analysis 49Q12, 65K10 |
| url | https://arxiv.org/abs/2510.15305 |