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

TL;DR

This paper introduces improved penalty-based bilevel optimization methods with a novel reformulation, a fully single-loop algorithm, and a flatness condition, leading to better convergence and efficiency in complex problems.

Contribution

It presents a new penalty reformulation for decoupling variables, a fully single-loop algorithm PBGD-Free, and a flatness condition that relaxes Lipschitz requirements, enhancing bilevel optimization efficiency.

Findings

Reduced iteration complexity for bilevel problems with coupled constraints.

Validated effectiveness through hyperparameter tuning in SVMs and language models.

Achieved larger step sizes and improved convergence rates.

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.