# Beyond backpropagation: bilevel optimization through implicit differentiation and equilibrium propagation

arXiv: https://arxiv.org/abs/2205.03076 · HTML: https://mlsanity.com/paper/2205.03076/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 104 references that are in the map.
- First posted on arXiv 6 May 2022.

## Builds on (most cited first, 30 of 43)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Meta-Learning with Latent Embedding Optimization](https://mlsanity.com/paper/1807.05960/) (2018)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Rapid Learning or Feature Reuse? Towards Understanding the Effectiveness of MAML](https://mlsanity.com/paper/1909.09157/) (2019)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (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)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functions](https://mlsanity.com/paper/1903.03088/) (2019)
- [Understanding and correcting pathologies in the training of learned optimizers](https://mlsanity.com/paper/1810.10180/) (2018)
- [Memory-Efficient Backpropagation Through Time](https://mlsanity.com/paper/1606.03401/) (2016)
- [Deep Declarative Networks: A New Hope](https://mlsanity.com/paper/1909.04866/) (2019)
- [Stochastic Hyperparameter Optimization through Hypernetworks](https://mlsanity.com/paper/1802.09419/) (2018)
- [A Game Theoretic Framework for Model Based Reinforcement Learning](https://mlsanity.com/paper/2004.07804/) (2020)
- [Scalable Gradient-Based Tuning of Continuous Regularization Hyperparameters](https://mlsanity.com/paper/1511.06727/) (2015)
- [Reviving and Improving Recurrent Back-Propagation](https://mlsanity.com/paper/1803.06396/) (2018)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)

## Built on by (most cited first, 26 of 26)

- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [Uncovering mesa-optimization algorithms in Transformers](https://mlsanity.com/paper/2309.05858/) (2023)
- [Energy-based learning algorithms for analog computing: a comparative study](https://mlsanity.com/paper/2312.15103/) (2023)
- [Brain-inspired Computational Intelligence via Predictive Coding](https://mlsanity.com/paper/2308.07870/) (2023)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Quantum Equilibrium Propagation: Gradient-Descent Training of Quantum Systems](https://mlsanity.com/paper/2406.00879/) (2024)
- [Improving equilibrium propagation without weight symmetry through Jacobian homeostasis](https://mlsanity.com/paper/2309.02214/) (2023)
- [Towards the Training of Deeper Predictive Coding Neural Networks](https://mlsanity.com/paper/2506.23800/) (2025)
- [Enhanced Meta Label Correction for Coping with Label Corruption](https://mlsanity.com/paper/2305.12961/) (2023)
- [On Optimal Regularization Parameters via Bilevel Learning](https://mlsanity.com/paper/2305.18394/) (2023)
- [A Review of Neuroscience-Inspired Machine Learning](https://mlsanity.com/paper/2403.18929/) (2024)
- [An adaptively inexact first-order method for bilevel optimization with application to hyperparameter learning](https://mlsanity.com/paper/2308.10098/) (2023)
- [A First-order Generative Bilevel Optimization Framework for Diffusion Models](https://mlsanity.com/paper/2502.08808/) (2025)
- [BiSSL: Enhancing the Alignment Between Self-Supervised Pretraining and Downstream Fine-Tuning via Bilevel Optimization](https://mlsanity.com/paper/2410.02387/) (2024)
- [A primal-dual algorithm for image reconstruction with input-convex neural network regularizers](https://mlsanity.com/paper/2410.12441/) (2024)
- [Efficient Morphology-Control Co-Design via Stackelberg Proximal Policy Optimization](https://mlsanity.com/paper/2603.15388/) (2026)
- [Robust Two-View Geometry Estimation with Implicit Differentiation](https://mlsanity.com/paper/2410.17983/) (2024)
- [Learning Set Functions with Implicit Differentiation](https://mlsanity.com/paper/2412.11239/) (2024)
- [BiRQ: Bi-Level Self-Labeling Random Quantization for Self-Supervised Speech Recognition](https://mlsanity.com/paper/2509.15430/) (2025)
- [Beyond sparse denoising in frames: minimax estimation with a scattering transform](https://mlsanity.com/paper/2510.19612/) (2025)
- [VertAX: a differentiable vertex model for learning epithelial tissue mechanics](https://mlsanity.com/paper/2604.06896/) (2026)
- [The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks](https://mlsanity.com/paper/2604.10272/) (2026)
- [Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machines](https://mlsanity.com/paper/2606.13454/) (2026)
- [Non-Linear Strategic Classification Made Practical](https://mlsanity.com/paper/2606.28204/) (2026)
- [A Light Bilevel Refinement Aligns Self-Supervised Representations for Stronger Task-Specific Learning](https://mlsanity.com/paper/2609.33424/) (2026)
- [BRIDGE: Bilevel Retrieval-Credit-Aware Agentic Reinforcement Learning](https://mlsanity.com/paper/2609.36505/) (2026)

## Abstract (arXiv)

This paper reviews gradient-based techniques to solve bilevel optimization problems. Bilevel optimization is a general way to frame the learning of systems that are implicitly defined through a quantity that they minimize. This characterization can be applied to neural networks, optimizers, algorithmic solvers and even physical systems, and allows for greater modeling flexibility compared to an explicit definition of such systems. Here we focus on gradient-based approaches that solve such problems. We distinguish them in two categories: those rooted in implicit differentiation, and those that leverage the equilibrium propagation theorem. We present the mathematical foundations that are behind such methods, introduce the gradient-estimation algorithms in detail and compare the competitive advantages of the different approaches.