Safe Gradient Flow for Bilevel Optimization

2025-01-27

AI Summary

This paper addresses key challenges in bilevel optimization—namely, the difficulty of satisfying lower-level constraints in real time and the high computational overhead of double-loop algorithms. We propose a control-theoretic single-loop framework. Our core contributions are: (1) a safety-gradient-flow dynamical system, where the upper-level optimization is formulated as a controlled gradient flow augmented with an embedded safety filter that enforces lower-level constraint feasibility in real time; (2) rigorous convergence analysis via Lyapunov stability theory, guaranteeing asymptotic convergence to a neighborhood of the optimal solution; and (3) a compact relaxation variant that efficiently handles high-dimensional lower-level problems by permitting bounded, controllable constraint violations. Experiments across multiple bilevel tasks demonstrate that our method significantly outperforms state-of-the-art double-loop and implicit-differentiation approaches, achieving superior computational efficiency and robustness.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Bilevel optimization is a key framework in hierarchical decision-making, where one problem is embedded within the constraints of another. In this work, we propose a control-theoretic approach to solving bilevel optimization problems. Our method consists of two components: a gradient flow mechanism to minimize the upper-level objective and a safety filter to enforce the constraints imposed by the lower-level problem. Together, these components form a safe gradient flow that solves the bilevel problem in a single loop. To improve scalability with respect to the lower-level problem's dimensions, we introduce a relaxed formulation and design a compact variant of the safe gradient flow. This variant minimizes the upper-level objective while ensuring the lower-level solution remains within a user-defined distance. Using Lyapunov analysis, we establish convergence guarantees for the dynamics, proving that they converge to a neighborhood of the optimal solution. Numerical experiments further validate the effectiveness of the proposed approaches. Our contributions provide both theoretical insights and practical tools for efficiently solving bilevel optimization problems.

Problem

• Bilevel Optimization
• Constraint Satisfaction
• Complex Problem Solving

Innovation

• Control Theory
• Gradient Flow
• Lyapunov Analysis