Beyond Backpropagation: Bilevel Optimization Through Implicit Differentiation and Equilibrium Propagation

Area of Science:

• Optimization
• Machine Learning
• Applied Mathematics

Background:

• Bilevel optimization extends statistical learning's loss minimization to implicitly defined systems.
• Such systems include neural networks, optimizers, and physical models, offering greater flexibility.
• Traditional methods often rely on explicit system definitions, limiting applicability.

Purpose of the Study:

• To review gradient-based techniques for solving bilevel optimization problems.
• To explore the application of implicit differentiation and equilibrium propagation in this domain.
• To provide a comprehensive comparison of different gradient estimation approaches.

Main Methods:

• Focuses on gradient descent for solving bilevel optimization.
• Utilizes the mathematical framework of implicit differentiation.
• Applies the equilibrium propagation theorem to this optimization setting.

Main Results:

• Presents the mathematical foundations for gradient-based bilevel optimization.
• Details gradient estimation algorithms for implicit systems.
• Compares the advantages of various bilevel optimization techniques.

Conclusions:

• Gradient-based methods, including implicit differentiation and equilibrium propagation, offer powerful tools for bilevel optimization.
• These techniques enhance modeling flexibility for complex systems like neural networks.
• The review provides a detailed algorithmic and comparative analysis for researchers.