# Functional Bilevel Optimization for Machine Learning

arXiv: https://arxiv.org/abs/2403.20233 · HTML: https://mlsanity.com/paper/2403.20233/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 98 references that are in the map.
- First posted on arXiv 29 Mar 2024.

## Builds on (most cited first, 30 of 57)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [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)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [SMASH: One-Shot Model Architecture Search through HyperNetworks](https://mlsanity.com/paper/1708.05344/) (2017)
- [Truncated Back-propagation for Bilevel Optimization](https://mlsanity.com/paper/1810.10667/) (2018)
- [Invariant Risk Minimization Games](https://mlsanity.com/paper/2002.04692/) (2020)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [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)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [Graph HyperNetworks for Neural Architecture Search](https://mlsanity.com/paper/1810.05749/) (2018)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization](https://mlsanity.com/paper/1607.05447/) (2016)
- [Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functions](https://mlsanity.com/paper/1903.03088/) (2019)
- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)

## Built on by (most cited first, 20 of 20)

- [Cautious Weight Decay](https://mlsanity.com/paper/2510.12402/) (2025)
- [Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal](https://mlsanity.com/paper/2411.19653/) (2024)
- [CoBo: Collaborative Learning via Bilevel Optimization](https://mlsanity.com/paper/2409.05539/) (2024)
- [Closed-Form Last Layer Optimization](https://mlsanity.com/paper/2510.04606/) (2025)
- [A First-order Generative Bilevel Optimization Framework for Diffusion Models](https://mlsanity.com/paper/2502.08808/) (2025)
- [Demystifying Spectral Feature Learning for Instrumental Variable Regression](https://mlsanity.com/paper/2506.10899/) (2025)
- [Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression](https://mlsanity.com/paper/2501.04898/) (2025)
- [A Fully First-Order Layer for Differentiable Optimization](https://mlsanity.com/paper/2512.02494/) (2025)
- [Learning Theory for Kernel Bilevel Optimization](https://mlsanity.com/paper/2502.08457/) (2025)
- [Outcome-Aware Spectral Feature Learning for Instrumental Variable Regression](https://mlsanity.com/paper/2512.00919/) (2025)
- [Non-Stationary Functional Bilevel Optimization](https://mlsanity.com/paper/2601.15363/) (2026)
- [Stackelberg Coupling of Online Representation Learning and Reinforcement Learning](https://mlsanity.com/paper/2508.07452/) (2025)
- [Bridging Constraints and Stochasticity: A Fully First-Order Method for Stochastic Bilevel Optimization with Linear Constraints](https://mlsanity.com/paper/2511.09845/) (2025)
- [Efficiency Follows Global-Local Decoupling](https://mlsanity.com/paper/2603.19567/) (2026)
- [Functional Natural Policy Gradients](https://mlsanity.com/paper/2603.28681/) (2026)
- [Functional Gradient Descent with Adaptive Representations](https://mlsanity.com/paper/2606.16926/) (2026)
- [AIFS-SUBS: Extending Data-Driven Forecasting to Sub-Seasonal Timescales](https://mlsanity.com/paper/2607.05100/) (2026)
- [From Data to Rewards: a Bilevel Optimization Perspective on Maximum Likelihood Estimation](https://mlsanity.com/paper/2510.07624/) (2025)
- [ROAD: Adaptive Data Mixing for Offline-to-Online Reinforcement Learning via Bi-Level Optimization](https://mlsanity.com/paper/2605.14497/) (2026)
- [Semiparametric Efficient Bilevel Gradient Estimation](https://mlsanity.com/paper/2605.21341/) (2026)

## Abstract (arXiv)

In this paper, we introduce a new functional point of view on bilevel optimization problems for machine learning, where the inner objective is minimized over a function space. These types of problems are most often solved by using methods developed in the parametric setting, where the inner objective is strongly convex with respect to the parameters of the prediction function. The functional point of view does not rely on this assumption and notably allows using over-parameterized neural networks as the inner prediction function. We propose scalable and efficient algorithms for the functional bilevel optimization problem and illustrate the benefits of our approach on instrumental regression and reinforcement learning tasks.