# Safe Gradient Flow for Bilevel Optimization

arXiv: https://arxiv.org/abs/2501.16520 · HTML: https://mlsanity.com/paper/2501.16520/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 52 references that are in the map.
- First posted on arXiv 27 Jan 2025.

## Builds on (most cited first, 25 of 25)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (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)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [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)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach](https://mlsanity.com/paper/2209.08709/) (2022)
- [Revisiting and Advancing Fast Adversarial Training Through The Lens of Bi-Level Optimization](https://mlsanity.com/paper/2112.12376/) (2021)
- [A Single-Timescale Method for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2102.04671/) (2021)
- [Efficient Gradient Approximation Method for Constrained Bilevel Optimization](https://mlsanity.com/paper/2302.01970/) (2023)
- [Stochastic Gradient Descent in Continuous Time](https://mlsanity.com/paper/1611.05545/) (2016)
- [Orthogonal Directions Constrained Gradient Method: from non-linear equality constraints to Stiefel manifold](https://mlsanity.com/paper/2303.09261/) (2023)
- [On Constraints in First-Order Optimization: A View from Non-Smooth Dynamical Systems](https://mlsanity.com/paper/2107.08225/) (2021)
- [First-Order Methods for Linearly Constrained Bilevel Optimization](https://mlsanity.com/paper/2406.12771/) (2024)
- [Prediction-Correction Interior-Point Method for Time-Varying Convex Optimization](https://mlsanity.com/paper/1608.07544/) (2016)
- [Control Barrier Function Based Design of Gradient Flows for Constrained Nonlinear Programming](https://mlsanity.com/paper/2204.01930/) (2022)
- [Two-Timescale Stochastic Gradient Descent in Continuous Time with Applications to Joint Online Parameter Estimation and Optimal Sensor Placement](https://mlsanity.com/paper/2007.15998/) (2020)
- [A Generalized Alternating Method for Bilevel Learning under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/2306.02422/) (2023)
- [An Inexact Conditional Gradient Method for Constrained Bilevel Optimization](https://mlsanity.com/paper/2306.02429/) (2023)
- [Zero-Order Control Barrier Functions for Sampled-Data Systems with State and Input Dependent Safety Constraints](https://mlsanity.com/paper/2411.17079/) (2024)

## Built on by (most cited first, 1 of 1)

- [Sequential QCQP for Bilevel Optimization with Line Search](https://mlsanity.com/paper/2505.14647/) (2025)

## Abstract (arXiv)

Bilevel optimization is a key framework in hierarchical decision-making, where one problem is embedded within the constraints of another. In this work, we propose a control-theoretic approach to solving bilevel optimization problems. Our method consists of two components: a gradient flow mechanism to minimize the upper-level objective and a safety filter to enforce the constraints imposed by the lower-level problem. Together, these components form a safe gradient flow that solves the bilevel problem in a single loop. To improve scalability with respect to the lower-level problem's dimensions, we introduce a relaxed formulation and design a compact variant of the safe gradient flow. This variant minimizes the upper-level objective while ensuring the lower-level decision variable remains within a user-defined suboptimality. Using Lyapunov analysis, we establish convergence guarantees for the dynamics, proving that they converge to a neighborhood of the optimal solution. Numerical experiments further validate the effectiveness of the proposed approaches. Our contributions provide both theoretical insights and practical tools for efficiently solving bilevel optimization problems.