mlsanityData to 7 Oct 2026

Beyond backpropagation: bilevel optimization through implicit differentiation and equilibrium propagation

arXiv 2205.03076 ↗First posted 6 May 2022cs.LG math.OCcited by 26

Not on the map. Why?

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.

Key facts. Cited by 26 papers in the map (data through 7 Oct 2026). Builds on 43 of its 104 references that are in the map. First posted on arXiv 6 May 2022.

Builds on

43 of 104 references.

References · 43

  1. 2020Language Models are Few-Shot Learnerscited by 32,199
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  3. 2022Generative Adversarial Networkscited by 5,603
  4. 2017Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networkscited by 4,861
  5. 2016HyperNetworkscited by 1,003
  6. 2019Deep Equilibrium Modelscited by 603
  7. 2017OptNet: Differentiable Optimization as a Layer in Neural Networkscited by 588
  8. 2019Meta-Learning with Implicit Gradientscited by 523
  9. 2018Meta-Learning with Latent Embedding Optimizationcited by 476
  10. 2019Meta-Learning with Differentiable Convex Optimizationcited by 461
  11. 2019Differentiable Convex Optimization Layerscited by 458
  12. 2018Meta-learning with differentiable closed-form solverscited by 395
31 more
  1. 2019Rapid Learning or Feature Reuse? Towards Understanding the Effectiveness of MAMLcited by 359
  2. 2019Optimizing Millions of Hyperparameters by Implicit Differentiationcited by 303
  3. 2016Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagationcited by 251
  4. 2016Hyperparameter optimization with approximate gradientcited by 231
  5. 2019Differentiation of Blackbox Combinatorial Solverscited by 224
  6. 2016Unrolled Generative Adversarial Networkscited by 208
  7. 2018Truncated Back-propagation for Bilevel Optimizationcited by 187
  8. 2019SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solvercited by 173
  9. 2020Multiscale Deep Equilibrium Modelscited by 149
  10. 2019Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functionscited by 104
  11. 2018Understanding and correcting pathologies in the training of learned optimizerscited by 97
  12. 2016Memory-Efficient Backpropagation Through Timecited by 78
  13. 2019Deep Declarative Networks: A New Hopecited by 77
  14. 2018Stochastic Hyperparameter Optimization through Hypernetworkscited by 72
  15. 2020A Game Theoretic Framework for Model Based Reinforcement Learningcited by 72
  16. 2015Scalable Gradient-Based Tuning of Continuous Regularization Hyperparameterscited by 71
  17. 2018Reviving and Improving Recurrent Back-Propagationcited by 69
  18. 2020Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Biascited by 60
  19. 2020Training End-to-End Analog Neural Networks with Equilibrium Propagationcited by 57
  20. 2016Connecting Generative Adversarial Networks and Actor-Critic Methodscited by 49
  21. 2017Unbiased Online Recurrent Optimizationcited by 46
  22. 2019A Unified Framework of Online Learning Algorithms for Training Recurrent Neural Networkscited by 37
  23. 2021Control-Oriented Model-Based Reinforcement Learning with Implicit Differentiationcited by 35
  24. 2018K for the Price of 1: Parameter-efficient Multi-task and Transfer Learningcited by 34
  25. 2017Equivalence of Equilibrium Propagation and Recurrent Backpropagationcited by 31
  26. 2015Training Deep Neural Networks via Direct Loss Minimizationcited by 25
  27. 2021A deep learning theory for neural networks grounded in physicscited by 23
  28. 2021Stackelberg Actor-Critic: Game-Theoretic Reinforcement Learning Algorithmscited by 18
  29. 2020Online Meta-Critic Learning for Off-Policy Actor-Critic Methodscited by 17
  30. 2021A contrastive rule for meta-learningcited by 14
  31. 2019On the Global Convergence of Actor-Critic: A Case for Linear Quadratic Regulator with Ergodic Costcited by 12

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 7; 2024: 6; 2025: 5; 2026: 7.

26 papers.

Sort:

By area:

  1. 2022Transformers learn in-context by gradient descentcited by 714
  2. 2023Uncovering mesa-optimization algorithms in Transformerscited by 94
  3. 2023Energy-based learning algorithms for analog computing: a comparative studycited by 36
  4. 2023Brain-inspired Computational Intelligence via Predictive Codingcited by 30
  5. 2025Learning Regularization Functionals for Inverse Problems: A Comparative Studycited by 12
  6. 2024Quantum Equilibrium Propagation: Gradient-Descent Training of Quantum Systemscited by 9
  7. 2023Improving equilibrium propagation without weight symmetry through Jacobian homeostasiscited by 8
  8. 2025Towards the Training of Deeper Predictive Coding Neural Networkscited by 7
  9. 2023Enhanced Meta Label Correction for Coping with Label Corruptioncited by 6
  10. 2024A Review of Neuroscience-Inspired Machine Learningcited by 6
  11. 2023On Optimal Regularization Parameters via Bilevel Learningcited by 4
  12. 2023An adaptively inexact first-order method for bilevel optimization with application to hyperparameter learningcited by 4
14 more
  1. 2025A First-order Generative Bilevel Optimization Framework for Diffusion Modelscited by 4
  2. 2024BiSSL: Enhancing the Alignment Between Self-Supervised Pretraining and Downstream Fine-Tuning via Bilevel Optimizationcited by 3
  3. 2024A primal-dual algorithm for image reconstruction with input-convex neural network regularizerscited by 2
  4. 2026Efficient Morphology-Control Co-Design via Stackelberg Proximal Policy Optimizationcited by 2
  5. 2024Robust Two-View Geometry Estimation with Implicit Differentiationcited by 1
  6. 2024Learning Set Functions with Implicit Differentiationcited by 1
  7. 2025BiRQ: Bi-Level Self-Labeling Random Quantization for Self-Supervised Speech Recognitioncited by 0
  8. 2025Beyond sparse denoising in frames: minimax estimation with a scattering transformcited by 0
  9. 2026VertAX: a differentiable vertex model for learning epithelial tissue mechanicscited by 0
  10. 2026The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networkscited by 0
  11. 2026Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machinescited by 0
  12. 2026Non-Linear Strategic Classification Made Practicalcited by 0
  13. 2026A Light Bilevel Refinement Aligns Self-Supervised Representations for Stronger Task-Specific Learningcited by 0
  14. 2026BRIDGE: Bilevel Retrieval-Credit-Aware Agentic Reinforcement Learningcited by 0