Riemannian Implicit Differentiation via a Fixed-Point Equation for Riemannian Bilevel Optimization

Area of Science:

• Optimization Theory
• Machine Learning
• Differential Geometry

Background:

• Riemannian meta-optimization (RMO) and metalearning are often bilevel optimization problems.
• Implicit differentiation effectively solves RMO by decoupling outer gradients but requires expert derivation for new tasks.
• Extending implicit differentiation to diverse Riemannian bilevel optimization tasks is challenging due to case-by-case derivation needs.

Purpose of the Study:

• To propose a unified Riemannian implicit differentiation method for flexible application across various Riemannian bilevel optimization tasks.
• To reduce the expert involvement typically required for deriving gradients in new optimization scenarios.
• To provide a general framework for solving Riemannian bilevel optimization problems.

Main Methods:

• Formulating the inner-level optimization as a root-finding process of a fixed-point equation for unified task representation.
• Deriving a unified expression for outer gradients by differentiating the fixed-point equation, thus avoiding task-specific derivations.
• Conducting convergence and approximation error analysis to validate the method's theoretical guarantees.

Main Results:

• A novel Riemannian implicit differentiation method with a unified expression for outer gradients is presented.
• The method demonstrates flexible application to various Riemannian optimization tasks with reduced expert involvement.
• Convergence and approximation error analyses confirm the method's effectiveness.

Conclusions:

• The proposed Riemannian implicit differentiation method offers a unified and flexible approach to solving bilevel optimization problems on Riemannian manifolds.
• This method significantly lowers the barrier for applying implicit differentiation to new and diverse Riemannian optimization tasks.
• Experimental validation confirms the method's effectiveness and broad applicability in the field.