A Single-Loop First-Order Algorithm for Linearly Constrained Bilevel Optimization

2025-10-28

AI Summary

This paper studies bilevel optimization with linear coupling constraints and a strongly convex lower-level problem, addressing challenges arising from nonsmoothness of the upper-level objective and costly Hessian computations. We propose SFLCB—a single-loop first-order algorithm—that reformulates the original bilevel problem into an equivalent single-level smooth optimization via penalty and augmented Lagrangian techniques. Crucially, we establish strong functional and gradient-level approximation guarantees between the reformulated objective and the original bilevel objective. Theoretically, SFLCB achieves an $mathcal{O}(varepsilon^{-3})$ convergence rate in terms of stationarity, improving upon the $mathcal{O}(varepsilon^{-3}log(varepsilon^{-1}))$ rate of conventional double-loop methods. Importantly, SFLCB relies solely on first-order information and avoids second-order computations entirely. Both theoretical analysis and empirical experiments demonstrate its superiority in convergence speed and practical efficiency. The implementation is publicly available.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study bilevel optimization problems where the lower-level problems are strongly convex and have coupled linear constraints. To overcome the potential non-smoothness of the hyper-objective and the computational challenges associated with the Hessian matrix, we utilize penalty and augmented Lagrangian methods to reformulate the original problem as a single-level one. Especially, we establish a strong theoretical connection between the reformulated function and the original hyper-objective by characterizing the closeness of their values and derivatives. Based on this reformulation, we propose a single-loop, first-order algorithm for linearly constrained bilevel optimization (SFLCB). We provide rigorous analyses of its non-asymptotic convergence rates, showing an improvement over prior double-loop algorithms -- form $O(epsilon^{-3}log(epsilon^{-1}))$ to $O(epsilon^{-3})$. The experiments corroborate our theoretical findings and demonstrate the practical efficiency of the proposed SFLCB algorithm. Simulation code is provided at https://github.com/ShenGroup/SFLCB.

Problem

• Solving linearly constrained bilevel optimization problems efficiently
• Overcoming non-smoothness and Hessian computational challenges
• Improving convergence rates for constrained bilevel optimization algorithms

Innovation

• Penalty and augmented Lagrangian methods reformulate problem
• Single-loop first-order algorithm reduces computational complexity
• Non-asymptotic convergence rate improves to O(ε^{-3})