A Framework for Bilevel Optimization on Riemannian Manifolds

Bilevel optimization has seen an increasing presence in various domains of applications. In this work, we propose a framework for solving bilevel optimization problems where variables of both lower and upper level problems are constrained on Riemannian manifolds. We provide several hypergradient estimation strategies on manifolds and study their estimation error. We provide convergence and complexity analysis for the proposed hypergradient descent algorithm on manifolds. We also extend the developments to stochastic bilevel optimization and to the use of general retraction. We showcase the utility of the proposed framework on various applications.