A Framework for Bilevel Optimization on Riemannian Manifolds
This work addresses bilevel optimization on manifolds, which is incremental as it extends existing methods to a specialized geometric setting.
The authors tackled bilevel optimization problems where variables are constrained on Riemannian manifolds by introducing a framework with hypergradient estimation strategies and convergence analysis, demonstrating efficacy through applications.
Bilevel optimization has gained prominence in various applications. In this study, we introduce a framework for solving bilevel optimization problems, where the variables in both the lower and upper levels are constrained on Riemannian manifolds. We present several hypergradient estimation strategies on manifolds and analyze their estimation errors. Furthermore, we provide comprehensive convergence and complexity analyses for the proposed hypergradient descent algorithm on manifolds. We also extend our framework to encompass stochastic bilevel optimization and incorporate the use of general retraction. The efficacy of the proposed framework is demonstrated through several applications.