# Value-Function-based Sequential Minimization for Bi-level Optimization

arXiv: https://arxiv.org/abs/2110.04974 · HTML: https://mlsanity.com/paper/2110.04974/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 64 references that are in the map.
- First posted on arXiv 11 Oct 2021.

## Builds on (most cited first, 27 of 27)

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Adversarial Attacks on Graph Neural Networks via Meta Learning](https://mlsanity.com/paper/1902.08412/) (2019)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Progressive Differentiable Architecture Search: Bridging the Depth Gap between Search and Evaluation](https://mlsanity.com/paper/1904.12760/) (2019)
- [Unrolled Generative Adversarial Networks](https://mlsanity.com/paper/1611.02163/) (2016)
- [Truncated Back-propagation for Bilevel Optimization](https://mlsanity.com/paper/1810.10667/) (2018)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)
- [Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond](https://mlsanity.com/paper/2101.11517/) (2021)
- [Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functions](https://mlsanity.com/paper/1903.03088/) (2019)
- [DARTS+: Improved Differentiable Architecture Search with Early Stopping](https://mlsanity.com/paper/1909.06035/) (2019)
- [Generalized Inner Loop Meta-Learning](https://mlsanity.com/paper/1910.01727/) (2019)
- [A Generic First-Order Algorithmic Framework for Bi-Level Programming Beyond Lower-Level Singleton](https://mlsanity.com/paper/2006.04045/) (2020)
- [A Value-Function-based Interior-point Method for Non-convex Bi-level Optimization](https://mlsanity.com/paper/2106.07991/) (2021)
- [Connecting Generative Adversarial Networks and Actor-Critic Methods](https://mlsanity.com/paper/1610.01945/) (2016)
- [Lower Bounds and Accelerated Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2102.03926/) (2021)
- [A General Descent Aggregation Framework for Gradient-based Bi-level Optimization](https://mlsanity.com/paper/2102.07976/) (2021)
- [Investigating Task-driven Latent Feasibility for Nonconvex Image Modeling](https://mlsanity.com/paper/1910.08242/) (2019)
- [On the Convergence of Learning-based Iterative Methods for Nonconvex Inverse Problems](https://mlsanity.com/paper/1808.05331/) (2018)
- [On ℓp-hyperparameter Learning via Bilevel Nonsmooth Optimization](https://mlsanity.com/paper/1806.01520/) (2018)

## Built on by (most cited first, 24 of 24)

- [BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach](https://mlsanity.com/paper/2209.08709/) (2022)
- [A Task-guided, Implicitly-searched and Meta-initialized Deep Model for Image Fusion](https://mlsanity.com/paper/2305.15862/) (2023)
- [An Introduction to Bi-level Optimization: Foundations and Applications in Signal Processing and Machine Learning](https://mlsanity.com/paper/2308.00788/) (2023)
- [Averaged Method of Multipliers for Bi-Level Optimization without Lower-Level Strong Convexity](https://mlsanity.com/paper/2302.03407/) (2023)
- [Functional Bilevel Optimization for Machine Learning](https://mlsanity.com/paper/2403.20233/) (2024)
- [Constrained Bi-Level Optimization: Proximal Lagrangian Value function Approach and Hessian-free Algorithm](https://mlsanity.com/paper/2401.16164/) (2024)
- [On Momentum-Based Gradient Methods for Bilevel Optimization with Nonconvex Lower-Level](https://mlsanity.com/paper/2303.03944/) (2023)
- [Optimal Hessian/Jacobian-Free Nonconvex-PL Bilevel Optimization](https://mlsanity.com/paper/2407.17823/) (2024)
- [A Primal-Dual-Assisted Penalty Approach to Bilevel Optimization with Coupled Constraints](https://mlsanity.com/paper/2406.10148/) (2024)
- [HUPE: Heuristic Underwater Perceptual Enhancement with Semantic Collaborative Learning](https://mlsanity.com/paper/2411.18296/) (2024)
- [Learning with Constraint Learning: New Perspective, Solution Strategy and Various Applications](https://mlsanity.com/paper/2307.15257/) (2023)
- [LancBiO: dynamic Lanczos-aided bilevel optimization via Krylov subspace](https://mlsanity.com/paper/2404.03331/) (2024)
- [Hierarchical Optimization-Derived Learning](https://mlsanity.com/paper/2302.05587/) (2023)
- [Efficient Curvature-Aware Hypergradient Approximation for Bilevel Optimization](https://mlsanity.com/paper/2505.02101/) (2025)
- [qNBO: quasi-Newton Meets Bilevel Optimization](https://mlsanity.com/paper/2502.01076/) (2025)
- [Learning Theory for Kernel Bilevel Optimization](https://mlsanity.com/paper/2502.08457/) (2025)
- [Non-Stationary Functional Bilevel Optimization](https://mlsanity.com/paper/2601.15363/) (2026)
- [Convex and Bilevel Optimization for Neuro-Symbolic Inference and Learning](https://mlsanity.com/paper/2401.09651/) (2024)
- [Faster Gradient Methods for Highly-Smooth Stochastic Bilevel Optimization](https://mlsanity.com/paper/2509.02937/) (2025)
- [Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems](https://mlsanity.com/paper/2605.08006/) (2026)
- [On Stability in Optimistic Bilevel Optimization](https://mlsanity.com/paper/2408.13323/) (2024)
- [Towards Differentiable Multilevel Optimization: A Gradient-Based Approach](https://mlsanity.com/paper/2410.11312/) (2024)
- [Sample Complexity Analysis for Constrained Bilevel Reinforcement Learning](https://mlsanity.com/paper/2602.00282/) (2026)
- [Efficient Bilevel Optimization for Meta Label Correction in Noisy Label Learning](https://mlsanity.com/paper/2605.17833/) (2026)

## Abstract (arXiv)

Gradient-based Bi-Level Optimization (BLO) methods have been widely applied to handle modern learning tasks. However, most existing strategies are theoretically designed based on restrictive assumptions (e.g., convexity of the lower-level sub-problem), and computationally not applicable for high-dimensional tasks. Moreover, there are almost no gradient-based methods able to solve BLO in those challenging scenarios, such as BLO with functional constraints and pessimistic BLO. In this work, by reformulating BLO into approximated single-level problems, we provide a new algorithm, named Bi-level Value-Function-based Sequential Minimization (BVFSM), to address the above issues. Specifically, BVFSM constructs a series of value-function-based approximations, and thus avoids repeated calculations of recurrent gradient and Hessian inverse required by existing approaches, time-consuming especially for high-dimensional tasks. We also extend BVFSM to address BLO with additional functional constraints. More importantly, BVFSM can be used for the challenging pessimistic BLO, which has never been properly solved before. In theory, we prove the asymptotic convergence of BVFSM on these types of BLO, in which the restrictive lower-level convexity assumption is discarded. To our best knowledge, this is the first gradient-based algorithm that can solve different kinds of BLO (e.g., optimistic, pessimistic, and with constraints) with solid convergence guarantees. Extensive experiments verify the theoretical investigations and demonstrate our superiority on various real-world applications.