Efficient Hessian-Free Methods for Multi-Objective Bilevel Optimization with Nonconvex Lower Level
Abstract
Multi-objective bilevel optimization has wide applications in the AI area such as automated learning and multi-task meta-learning. Although recently some works have been begun to study the multi-objective bilevel optimization, the proposed methods rely on the (strongly) convex lower level problems. In fact, these multi-objective bilevel learning problems are generally nonconvex, and particularly their lower level problems are nonconvex. To fill this gap, we propose a class of Multi-Objective Moreau Envelope based Hessian-free Algorithms (MOMEHA) for the multi-objective bilevel learning problems with nonconvex lower level. Specifically, our method uses the Moreau envelope to relax the original problem into a multi-objective single-level optimization with an envelope constraint. In particular, our method retains computational advantages of being single-loop and Hessian-free in the multi-objective setting by incorporating a smooth weighted Tchebycheff scalarization. Furthermore, we propose a momentum-based variant of MOMEHA (i.e., MB-MOMEHA) method for the stochastic multi-objective bilevel learning problems. In theory, we provide the convergence properties of our algorithms under both deterministic and stochastic setting. Some experiments on few-shot meta-learning and neural architecture search demonstrate that our methods outperform the existing approaches in Pareto front, validating its effectiveness and robustness.
Related
- Matching Lower Bounds for Randomized First-Order Methods in Hessian-Lipschitz Nonconvex Optimization
- Matching Full-Hessian Oracle Complexity for Nonconvex--Strongly-Concave Minimax Optimization
- Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization
- Inertial Primal-Dual Dynamics Methods Featuring Implicit Hessian-Driven Damping for Convex Optimization Problems in Continuous and Discrete Time
- Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems
BibTeX
@misc{jiang2026efficient,
title = {{Efficient Hessian-Free Methods for Multi-Objective Bilevel Optimization with Nonconvex Lower Level}},
author = {Yicong Jiang and Feihu Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.12704},
eprint = {2608.12704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12704},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Efficient Hessian-Free Methods for Multi-Objective Bilevel Optimization with Nonconvex Lower Level
AU - Yicong Jiang
AU - Feihu Huang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2608.12704
UR - https://arxiv.org/abs/2608.12704
AB - Multi-objective bilevel optimization has wide applications in the AI area such as automated learning and multi-task meta-learning. Although recently some works have been begun to study the multi-objective bilevel optimization, the proposed methods rely on the (strongly) convex lower level problems. In fact, these multi-objective bilevel learning problems are generally nonconvex, and particularly their lower level problems are nonconvex. To fill this gap, we propose a class of Multi-Objective Moreau Envelope based Hessian-free Algorithms (MOMEHA) for the multi-objective bilevel learning problems with nonconvex lower level. Specifically, our method uses the Moreau envelope to relax the original problem into a multi-objective single-level optimization with an envelope constraint. In particular, our method retains computational advantages of being single-loop and Hessian-free in the multi-objective setting by incorporating a smooth weighted Tchebycheff scalarization. Furthermore, we propose a momentum-based variant of MOMEHA (i.e., MB-MOMEHA) method for the stochastic multi-objective bilevel learning problems. In theory, we provide the convergence properties of our algorithms under both deterministic and stochastic setting. Some experiments on few-shot meta-learning and neural architecture search demonstrate that our methods outperform the existing approaches in Pareto front, validating its effectiveness and robustness.
ER -