Efficient Penalty-Based Bilevel Methods: Improved Analysis, Novel Updates, and Flatness Condition

• URL: http://arxiv.org/abs/2511.16796v1
• Date: Thu, 20 Nov 2025 20:48:14 GMT
• Title: Efficient Penalty-Based Bilevel Methods: Improved Analysis, Novel Updates, and Flatness Condition
• Authors: Liuyuan Jiang, Quan Xiao, Lisha Chen, Tianyi Chen,
• Abstract summary: Penalty-based methods have become popular for solving bilevel optimization (BLO) problems.<n>They often require inner-loop iteration to solve the lower-level (LL) problem and small outer-loop step sizes to handle the increased smoothness induced by large penalty terms.<n>This work considers the general BLO problems with coupled constraints (CCs) and leverages a novel penalty reformulation that decouples the upper- and lower-level variables.
• Score: 51.22672287601796
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Penalty-based methods have become popular for solving bilevel optimization (BLO) problems, thanks to their effective first-order nature. However, they often require inner-loop iterations to solve the lower-level (LL) problem and small outer-loop step sizes to handle the increased smoothness induced by large penalty terms, leading to suboptimal complexity. This work considers the general BLO problems with coupled constraints (CCs) and leverages a novel penalty reformulation that decouples the upper- and lower-level variables. This yields an improved analysis of the smoothness constant, enabling larger step sizes and reduced iteration complexity for Penalty-Based Gradient Descent algorithms in ALTernating fashion (ALT-PBGD). Building on the insight of reduced smoothness, we propose PBGD-Free, a novel fully single-loop algorithm that avoids inner loops for the uncoupled constraint BLO. For BLO with CCs, PBGD-Free employs an efficient inner-loop with substantially reduced iteration complexity. Furthermore, we propose a novel curvature condition describing the "flatness" of the upper-level objective with respect to the LL variable. This condition relaxes the traditional upper-level Lipschitz requirement, enables smaller penalty constant choices, and results in a negligible penalty gradient term during upper-level variable updates. We provide rigorous convergence analysis and validate the method's efficacy through hyperparameter optimization for support vector machines and fine-tuning of large language models.

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