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    This study introduces a unified Riemannian implicit differentiation method for bilevel optimization problems. It simplifies complex derivations, enabling broader application in Riemannian meta-optimization and metalearning tasks.

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    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.