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    This study introduces a novel gradient-based algorithm for Bi-Level Optimization (BLO), overcoming limitations of existing methods. The new approach, Bi-level Value-Function-based Sequential Minimization (BVFSM), efficiently handles high-dimensional and constrained BLO problems, including pessimistic scenarios.

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    Area of Science:

    • Machine Learning
    • Optimization Theory
    • Computational Mathematics

    Background:

    • Gradient-based Bi-Level Optimization (BLO) methods are crucial for modern learning tasks but face limitations.
    • Existing methods often rely on restrictive assumptions like lower-level convexity and struggle with high-dimensional problems.
    • Current gradient-based approaches are inadequate for complex BLO scenarios, including functional constraints and pessimistic BLO.

    Purpose of the Study:

    • To develop a novel gradient-based algorithm for Bi-Level Optimization (BLO) that addresses computational and theoretical limitations of existing methods.
    • To create an efficient algorithm applicable to high-dimensional BLO, including problems with functional constraints and pessimistic formulations.
    • To provide theoretical convergence guarantees for the proposed algorithm across various BLO types.

    Main Methods:

    • Reformulating BLO into approximated single-level problems.
    • Developing a Bi-level Value-Function-based Sequential Minimization (BVFSM) algorithm utilizing value-function approximations.
    • Extending BVFSM to incorporate functional constraints and handle pessimistic BLO.

    Main Results:

    • BVFSM avoids computationally expensive recurrent gradient and Hessian inverse calculations, improving efficiency for high-dimensional tasks.
    • The algorithm is successfully extended to address BLO with functional constraints.
    • BVFSM demonstrates capability in solving pessimistic BLO, a previously unsolved challenge, with proven asymptotic convergence without lower-level convexity assumptions.

    Conclusions:

    • BVFSM is the first gradient-based algorithm to solve optimistic, pessimistic, and constrained BLO with convergence guarantees.
    • The method offers a significant advancement for tackling complex and high-dimensional BLO problems.
    • Extensive experiments validate BVFSM's theoretical findings and demonstrate its superior performance in real-world applications.