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A Tale of Two Problems: Multi-Objective Bilevel Learning Meets Equality Constrained Multi-Objective Optimization

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 5.5Download Report PDF
Multi-objective optimizationBilevel optimizationPreference.
Abstract:

In recent years, bilevel optimization (BLO) has attracted significant attention for its broad applications in machine learning. However, most existing works on BLO remain confined to the single-objective setting and rely on the lower-level strong convexity assumption, which significantly restricts their applicability to modern machine learning problems of growing complexity. In this paper, we make the first attempt to extend BLO to the multi-objective setting under a relaxed lower-level general convexity (LLGC) assumption. To this end, we reformulate the multi-objective bilevel learning (MOBL) problem with LLGC into an equality constrained multi-objective optimization (ECMO) problem. This transformation yields a single-level formulation that is more amenable to algorithm design while preserving the optimal solutions of the original MOBL problem. However, ECMO itself is a new problem that has not yet been studied in the literature, with no existing results on its algorithmic design or theoretical analysis, and without a formally established convergence metric. To address this gap, we first establish a new Karush–Kuhn–Tucker (KKT)-based Pareto stationarity as the convergence criterion for ECMO algorithm design. Based on this foundation, we propose a weighted Chebyshev (WC)-penalty algorithm that achieves a finite-time convergence rate of O(ST−12)\mathcal{O}(ST^{-\frac{1}{2}}) to KKT-based Pareto stationarity in both deterministic and stochastic settings, where SS denotes the number of objectives, and TT is the total iterations. Moreover, by varying the preference vector over the SS-dimensional simplex, our WC-penalty method systematically explores the Pareto front. Finally, solutions to the ECMO problem translate directly into solutions for the original MOBL problem, thereby closing the loop between these two foundational optimization frameworks. We verify the efficacy of our approach through experiments on multi-objective data weighting in reinforcement learning from human feedback (RLHF) reward model training and large language model (LLM) alignment.

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Overview

Overall Novelty Assessment

The paper reformulates multi-objective bilevel learning under lower-level general convexity (LLGC-MOBL) as an equality-constrained multi-objective optimization (ECMO) problem and proposes a KKT-based Pareto stationarity criterion with a WC-Penalty algorithm. It resides in the Multi-Objective Bilevel Optimization Theory leaf alongside three sibling papers. This leaf is relatively sparse within the broader taxonomy of 25 papers across multiple branches, indicating that multi-objective bilevel theory under relaxed convexity assumptions remains an emerging research direction rather than a crowded subfield.

The taxonomy reveals neighboring leaves addressing single-objective bilevel methods under general convexity (two papers) and stochastic or decomposition-based approaches. The paper's focus on multi-objective formulations and general convexity distinguishes it from single-objective siblings and from strongly convex lower-level methods. The scope note for its leaf explicitly excludes single-objective and purely single-level multi-objective work, positioning this contribution at the intersection of multi-criteria decision-making and relaxed convexity assumptions—a boundary less explored than traditional strongly convex bilevel settings.

Among 28 candidates examined, the reformulation contribution (Contribution A) encountered one refutable candidate out of ten examined, suggesting some prior work on problem transformation. The KKT-based stationarity criterion (Contribution B) and WC-Penalty algorithm (Contribution C) showed no refutable candidates among ten and eight examined, respectively. This limited search scope indicates that while the ECMO reformulation may overlap with existing techniques, the convergence metric and algorithmic framework appear less directly anticipated in the top-30 semantic matches, though exhaustive coverage cannot be claimed.

Based on the top-28 semantic matches and taxonomy structure, the work appears to occupy a relatively underexplored niche—multi-objective bilevel optimization without strong convexity. The analysis does not cover the full breadth of optimization literature, and the single refutable pair for the reformulation suggests potential incremental overlap in problem transformation techniques. The algorithmic and theoretical contributions show fewer direct precedents within the examined scope, though broader searches might reveal additional related work.

Taxonomy

Core-task Taxonomy Papers
25
3
Claimed Contributions
28
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: multi-objective bilevel optimization under lower-level general convexity. This field addresses hierarchical decision-making problems where a leader optimizes multiple objectives subject to a follower's response, and the follower's problem exhibits general convex structure. The taxonomy organizes the literature into four main branches. Theoretical Foundations and Algorithmic Frameworks (including works like Multiobjective Bilevel Survey[6] and Bilevel Optimization[4]) establish core concepts, optimality conditions, and general-purpose algorithmic schemes. Specialized Problem Structures and Extensions explore variants such as robust formulations (Robust Minmax Bilevel[2]), semi-vectorial settings (Semi Vectorial Bilevel[8]), and discrete or mixed-integer cases (Discrete Complexity[14], Mixed Integer Convex[9]). Solution Methodologies develop concrete techniques ranging from decomposition methods (Benders Decomposition[13]) to evolutionary and hybrid approaches (Coevolutionary Hybrid[16], Fuzzy Stochastic Genetic[12]). Application Domains demonstrate the framework's utility in areas like network design (Dense Wireless Networks[25]) and product family optimization (Stackelberg Product Family[17]). Recent activity highlights a tension between theoretical rigor and computational tractability. Many studies pursue exact or near-exact solution concepts under strong convexity assumptions (Convex Lower Level[1]), while others embrace heuristic or stochastic methods to handle real-world complexity (Stochastic Bilevel Algorithms[10], Two Timescale Actor Critic[3]). Two Problems Bilevel[0] sits within the Theoretical Foundations branch, focusing on multi-objective bilevel optimization theory. Its emphasis on lower-level general convexity positions it close to foundational works like Multiobjective Bilevel Survey[6] and Semi Vectorial Bilevel[8], yet it appears to address dual problem instances or coupled objectives more explicitly than these neighbors. This contrasts with purely algorithmic contributions and suggests a focus on characterizing solution sets and optimality under relaxed convexity, bridging classical theory and emerging stochastic or robust extensions.

Claimed Contributions

Reformulation of LLGC-MOBL as ECMO problem

The authors transform the multi-objective bilevel learning problem under lower-level general convexity assumption into a single-level equality constrained multi-objective optimization problem. This reformulation preserves optimal solutions while making the problem more amenable to algorithm design.

10 retrieved papers
Can Refute
KKT-based Pareto stationarity for ECMO

The authors establish a novel characterization of Pareto stationarity for equality constrained multi-objective optimization using the KKT conditions of the weighted-Chebyshev scalarized problem. This provides the first formally established convergence metric for ECMO problems.

10 retrieved papers
WC-Penalty algorithm with convergence guarantees

The authors develop a weighted Chebyshev penalty algorithm for solving ECMO problems with proven finite-time convergence rate guarantees. The algorithm systematically explores the Pareto front by varying preference vectors over the simplex.

8 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Reformulation of LLGC-MOBL as ECMO problem

The authors transform the multi-objective bilevel learning problem under lower-level general convexity assumption into a single-level equality constrained multi-objective optimization problem. This reformulation preserves optimal solutions while making the problem more amenable to algorithm design.

Contribution

KKT-based Pareto stationarity for ECMO

The authors establish a novel characterization of Pareto stationarity for equality constrained multi-objective optimization using the KKT conditions of the weighted-Chebyshev scalarized problem. This provides the first formally established convergence metric for ECMO problems.

Contribution

WC-Penalty algorithm with convergence guarantees

The authors develop a weighted Chebyshev penalty algorithm for solving ECMO problems with proven finite-time convergence rate guarantees. The algorithm systematically explores the Pareto front by varying preference vectors over the simplex.

Table of Contents