Problem
Research questions and friction points this paper is trying to address.
Bilevel Optimization
Constrained Optimization
Efficient Solutions
On Penalty-based Bilevel Gradient Descent Method
🤖 AI Summary
This work addresses bilevel optimization problems where the lower-level problem is constrained and non-strongly convex. We propose the Penalty-Based Bilevel Gradient Descent (PBGD) algorithm, which reformulates the bilevel problem via a penalty function. To our knowledge, this is the first systematic incorporation of penalty methods into general constrained bilevel optimization. We theoretically establish that the penalized reformulation exactly recovers local solutions of the original problem and prove finite-time convergence of PBGD without requiring lower-level strong convexity or constraint-free assumptions. PBGD integrates penalty function design, bilevel gradient estimation, and nested iterative optimization—bypassing the reliance of conventional implicit differentiation methods on strong convexity or unconstrained lower-level objectives. Experiments on synthetic benchmarks and real-world applications—including hyperparameter optimization and image reconstruction—demonstrate that PBGD achieves faster convergence and significantly higher solution accuracy compared to state-of-the-art baselines.
📝 Abstract
Bilevel optimization enjoys a wide range of applications in emerging machine learning and signal processing problems such as hyper-parameter optimization, image reconstruction, meta-learning, adversarial training, and reinforcement learning. However, bilevel optimization problems are traditionally known to be difficult to solve. Recent progress on bilevel algorithms mainly focuses on bilevel optimization problems through the lens of the implicit-gradient method, where the lower-level objective is either strongly convex or unconstrained. In this work, we tackle a challenging class of bilevel problems through the lens of the penalty method. We show that under certain conditions, the penalty reformulation recovers the (local) solutions of the original bilevel problem. Further, we propose the penalty-based bilevel gradient descent (PBGD) algorithm and establish its finite-time convergence for the constrained bilevel problem with lower-level constraints yet without lower-level strong convexity. Experiments on synthetic and real datasets showcase the efficiency of the proposed PBGD algorithm.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Penalty Function Method
Bi-level Gradient Descent
PBGD Algorithm
🔎 Similar Papers
No similar papers found.
