# Credit Assignment in Neural Networks through Deep Feedback Control

arXiv: https://arxiv.org/abs/2106.07887 · HTML: https://mlsanity.com/paper/2106.07887/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 78 references that are in the map.
- First posted on arXiv 15 Jun 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Tune: A Research Platform for Distributed Model Selection and Training](https://mlsanity.com/paper/1807.05118/) (2018)
- [Direct Feedback Alignment Provides Learning in Deep Neural Networks](https://mlsanity.com/paper/1609.01596/) (2016)
- [Difference Target Propagation](https://mlsanity.com/paper/1412.7525/) (2014)
- [Practical Gauss-Newton Optimisation for Deep Learning](https://mlsanity.com/paper/1706.03662/) (2017)
- [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)
- [Deep Learning without Weight Transport](https://mlsanity.com/paper/1904.05391/) (2019)
- [How Auto-Encoders Could Provide Credit Assignment in Deep Networks via Target Propagation](https://mlsanity.com/paper/1407.7906/) (2014)
- [Fast Convergence of Natural Gradient Descent for Overparameterized Neural Networks](https://mlsanity.com/paper/1905.10961/) (2019)
- [Towards deep learning with segregated dendrites](https://mlsanity.com/paper/1610.00161/) (2016)
- [Towards Biologically Plausible Deep Learning](https://mlsanity.com/paper/1502.04156/) (2015)
- [A Theoretical Framework for Target Propagation](https://mlsanity.com/paper/2006.14331/) (2020)
- [Gram-Gauss-Newton Method: Learning Overparameterized Neural Networks for Regression Problems](https://mlsanity.com/paper/1905.11675/) (2019)
- [Feedback alignment in deep convolutional networks](https://mlsanity.com/paper/1812.06488/) (2018)
- [Learning to solve the credit assignment problem](https://mlsanity.com/paper/1906.00889/) (2019)
- [Direct Feedback Alignment with Sparse Connections for Local Learning](https://mlsanity.com/paper/1903.02083/) (2019)
- [Principled Training of Neural Networks with Direct Feedback Alignment](https://mlsanity.com/paper/1906.04554/) (2019)
- [Two Routes to Scalable Credit Assignment without Weight Symmetry](https://mlsanity.com/paper/2003.01513/) (2020)
- [Error Forward-Propagation: Reusing Feedforward Connections to Propagate Errors in Deep Learning](https://mlsanity.com/paper/1808.03357/) (2018)
- [Deriving Differential Target Propagation from Iterating Approximate Inverses](https://mlsanity.com/paper/2007.15139/) (2020)
- [Predicting non-linear dynamics by stable local learning in a recurrent spiking neural network](https://mlsanity.com/paper/1702.06463/) (2017)
- [The brain as an efficient and robust adaptive learner](https://mlsanity.com/paper/1705.08031/) (2017)
- [Biological credit assignment through dynamic inversion of feedforward networks](https://mlsanity.com/paper/2007.05112/) (2020)
- [Spike-based causal inference for weight alignment](https://mlsanity.com/paper/1910.01689/) (2019)
- [Learning arbitrary dynamics in efficient, balanced spiking networks using local plasticity rules](https://mlsanity.com/paper/1705.08026/) (2017)
- [A biologically plausible neural network for local supervision in cortical microcircuits](https://mlsanity.com/paper/2011.15031/) (2020)

## Built on by (most cited first, 29 of 29)

- [A Theoretical Framework for Inference and Learning in Predictive Coding Networks](https://mlsanity.com/paper/2207.12316/) (2022)
- [The least-control principle for local learning at equilibrium](https://mlsanity.com/paper/2207.01332/) (2022)
- [Forward Learning with Top-Down Feedback: Empirical and Analytical Characterization](https://mlsanity.com/paper/2302.05440/) (2023)
- [Minimizing Control for Credit Assignment with Strong Feedback](https://mlsanity.com/paper/2204.07249/) (2022)
- [Improving equilibrium propagation without weight symmetry through Jacobian homeostasis](https://mlsanity.com/paper/2309.02214/) (2023)
- [Learning efficient backprojections across cortical hierarchies in real time](https://mlsanity.com/paper/2212.10249/) (2022)
- [Bio-Inspired, Task-Free Continual Learning through Activity Regularization](https://mlsanity.com/paper/2212.04316/) (2022)
- [Counter-Current Learning: A Biologically Plausible Dual Network Approach for Deep Learning](https://mlsanity.com/paper/2409.19841/) (2024)
- [Thalamus: a brain-inspired algorithm for biologically-plausible continual learning and disentangled representations](https://mlsanity.com/paper/2205.11713/) (2022)
- [Target Propagation via Regularized Inversion](https://mlsanity.com/paper/2112.01453/) (2021)
- [Beyond accuracy: generalization properties of bio-plausible temporal credit assignment rules](https://mlsanity.com/paper/2206.00823/) (2022)
- [LightFF: Lightweight Inference for Forward-Forward Algorithm](https://mlsanity.com/paper/2404.05241/) (2024)
- [Duality Principle and Biologically Plausible Learning: Connecting the Representer Theorem and Hebbian Learning](https://mlsanity.com/paper/2309.16687/) (2023)
- [Toward Practical Equilibrium Propagation: Brain-inspired Recurrent Neural Network with Feedback Regulation and Residual Connections](https://mlsanity.com/paper/2508.11659/) (2025)
- [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)
- [Intrinsic Biologically Plausible Adversarial Robustness](https://mlsanity.com/paper/2309.17348/) (2023)
- [A Unified Framework for Neural Computation and Learning Over Time](https://mlsanity.com/paper/2409.12038/) (2024)
- [Spike-based alignment learning solves the weight transport problem](https://mlsanity.com/paper/2503.02642/) (2025)
- [Credit Assignment via Neural Manifold Noise Correlation](https://mlsanity.com/paper/2601.02636/) (2026)
- [Self-supervised local learning rules learn the hidden hierarchical structure of high-dimensional data](https://mlsanity.com/paper/2605.18557/) (2026)
- [Biologically Plausible Training of Deep Neural Networks Using a Top-down Credit Assignment Network](https://mlsanity.com/paper/2208.01416/) (2022)
- [Dis-inhibitory neuronal circuits can control the sign of synaptic plasticity](https://mlsanity.com/paper/2310.19614/) (2023)
- [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)
- [Mixed-signal implementation of feedback-control optimizer for single-layer Spiking Neural Networks](https://mlsanity.com/paper/2603.24113/) (2026)
- [Graph-based Target Back-Propagation for Context Adaptation in Multi-LLM Agentic Systems](https://mlsanity.com/paper/2606.14155/) (2026)
- [The Ups and Downs of Backprop Weights](https://mlsanity.com/paper/2609.22554/) (2026)
- [Gated Target Propagation for Compositional Generalization in Continual Learning](https://mlsanity.com/paper/2610.04649/) (2026)

## Abstract (arXiv)

The success of deep learning sparked interest in whether the brain learns by using similar techniques for assigning credit to each synaptic weight for its contribution to the network output. However, the majority of current attempts at biologically-plausible learning methods are either non-local in time, require highly specific connectivity motives, or have no clear link to any known mathematical optimization method. Here, we introduce Deep Feedback Control (DFC), a new learning method that uses a feedback controller to drive a deep neural network to match a desired output target and whose control signal can be used for credit assignment. The resulting learning rule is fully local in space and time and approximates Gauss-Newton optimization for a wide range of feedback connectivity patterns. To further underline its biological plausibility, we relate DFC to a multi-compartment model of cortical pyramidal neurons with a local voltage-dependent synaptic plasticity rule, consistent with recent theories of dendritic processing. By combining dynamical system theory with mathematical optimization theory, we provide a strong theoretical foundation for DFC that we corroborate with detailed results on toy experiments and standard computer-vision benchmarks.