# The least-control principle for local learning at equilibrium

arXiv: https://arxiv.org/abs/2207.01332 · HTML: https://mlsanity.com/paper/2207.01332/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 108 references that are in the map.
- First posted on arXiv 4 Jul 2022.

## Builds on (most cited first, 30 of 40)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [On First-Order Meta-Learning Algorithms](https://mlsanity.com/paper/1803.02999/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (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)
- [Direct Feedback Alignment Provides Learning in Deep Neural Networks](https://mlsanity.com/paper/1609.01596/) (2016)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Difference Target Propagation](https://mlsanity.com/paper/1412.7525/) (2014)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Assessing the Scalability of Biologically-Motivated Deep Learning Algorithms and Architectures](https://mlsanity.com/paper/1807.04587/) (2018)
- [Dendritic cortical microcircuits approximate the backpropagation algorithm](https://mlsanity.com/paper/1810.11393/) (2018)
- [Brain-Like Object Recognition with High-Performing Shallow Recurrent ANNs](https://mlsanity.com/paper/1909.06161/) (2019)
- [Deep Learning without Weight Transport](https://mlsanity.com/paper/1904.05391/) (2019)
- [Deep Declarative Networks: A New Hope](https://mlsanity.com/paper/1909.04866/) (2019)
- [Scalable Gradient-Based Tuning of Continuous Regularization Hyperparameters](https://mlsanity.com/paper/1511.06727/) (2015)
- [Towards deep learning with segregated dendrites](https://mlsanity.com/paper/1610.00161/) (2016)
- [Reviving and Improving Recurrent Back-Propagation](https://mlsanity.com/paper/1803.06396/) (2018)
- [A Theoretical Framework for Target Propagation](https://mlsanity.com/paper/2006.14331/) (2020)
- [Bridging the Gaps Between Residual Learning, Recurrent Neural Networks and Visual Cortex](https://mlsanity.com/paper/1604.03640/) (2016)
- [Meta-learning in natural and artificial intelligence](https://mlsanity.com/paper/2011.13464/) (2020)

## Built on by (most cited first, 16 of 16)

- [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)
- [Online learning of long-range dependencies](https://mlsanity.com/paper/2305.15947/) (2023)
- [How connectivity structure shapes rich and lazy learning in neural circuits](https://mlsanity.com/paper/2310.08513/) (2023)
- [Improving equilibrium propagation without weight symmetry through Jacobian homeostasis](https://mlsanity.com/paper/2309.02214/) (2023)
- [ePC: Fast and Deep Predictive Coding in Digital Simulation](https://mlsanity.com/paper/2505.20137/) (2025)
- [Flexible Phase Dynamics for Bio-Plausible Contrastive Learning](https://mlsanity.com/paper/2302.12431/) (2023)
- [Duality Principle and Biologically Plausible Learning: Connecting the Representer Theorem and Hebbian Learning](https://mlsanity.com/paper/2309.16687/) (2023)
- [Continual Learning through Control Minimization](https://mlsanity.com/paper/2602.04542/) (2026)
- [A feedback control optimizer for online and hardware-aware training of Spiking Neural Networks](https://mlsanity.com/paper/2602.13261/) (2026)
- [Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example](https://mlsanity.com/paper/2506.06904/) (2025)
- [Credit Assignment via Neural Manifold Noise Correlation](https://mlsanity.com/paper/2601.02636/) (2026)
- [Nature-Inspired Local Propagation](https://mlsanity.com/paper/2402.05959/) (2024)
- [Contribute to balance, wire in accordance: Emergence of backpropagation from a simple, bio-plausible neuroplasticity rule](https://mlsanity.com/paper/2405.14139/) (2024)
- [Stimulus-to-Stimulus Learning in RNNs with Cortical Inductive Biases](https://mlsanity.com/paper/2409.13471/) (2024)
- [Generalising E-prop to Deep Networks](https://mlsanity.com/paper/2512.24506/) (2025)

## Abstract (arXiv)

Equilibrium systems are a powerful way to express neural computations. As special cases, they include models of great current interest in both neuroscience and machine learning, such as deep neural networks, equilibrium recurrent neural networks, deep equilibrium models, or meta-learning. Here, we present a new principle for learning such systems with a temporally- and spatially-local rule. Our principle casts learning as a least-control problem, where we first introduce an optimal controller to lead the system towards a solution state, and then define learning as reducing the amount of control needed to reach such a state. We show that incorporating learning signals within a dynamics as an optimal control enables transmitting activity-dependent credit assignment information, avoids storing intermediate states in memory, and does not rely on infinitesimal learning signals. In practice, our principle leads to strong performance matching that of leading gradient-based learning methods when applied to an array of problems involving recurrent neural networks and meta-learning. Our results shed light on how the brain might learn and offer new ways of approaching a broad class of machine learning problems.