# Predictive Coding Approximates Backprop along Arbitrary Computation Graphs

arXiv: https://arxiv.org/abs/2006.04182 · HTML: https://mlsanity.com/paper/2006.04182/

## Key facts

- Cited by 90 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2020.

## Built on by (most cited first, 30 of 90)

- [Predictive Coding: a Theoretical and Experimental Review](https://mlsanity.com/paper/2107.12979/) (2021)
- [Hebbian Deep Learning Without Feedback](https://mlsanity.com/paper/2209.11883/) (2022)
- [Predictive Coding: Towards a Future of Deep Learning beyond Backpropagation?](https://mlsanity.com/paper/2202.09467/) (2022)
- [Associative Memories via Predictive Coding](https://mlsanity.com/paper/2109.08063/) (2021)
- [Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations](https://mlsanity.com/paper/2209.00530/) (2022)
- [Learning on Arbitrary Graph Topologies via Predictive Coding](https://mlsanity.com/paper/2201.13180/) (2022)
- [SoftHebb: Bayesian Inference in Unsupervised Hebbian Soft Winner-Take-All Networks](https://mlsanity.com/paper/2107.05747/) (2021)
- [Brain-inspired Computational Intelligence via Predictive Coding](https://mlsanity.com/paper/2308.07870/) (2023)
- [Hybrid Predictive Coding: Inferring, Fast and Slow](https://mlsanity.com/paper/2204.02169/) (2022)
- [Backpropagation at the Infinitesimal Inference Limit of Energy-Based Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive Hebbian Learning](https://mlsanity.com/paper/2206.02629/) (2022)
- [Reverse Differentiation via Predictive Coding](https://mlsanity.com/paper/2103.04689/) (2021)
- [A Stable, Fast, and Fully Automatic Learning Algorithm for Predictive Coding Networks](https://mlsanity.com/paper/2212.00720/) (2022)
- [On the relationship between predictive coding and backpropagation](https://mlsanity.com/paper/2106.13082/) (2021)
- [A Theoretical Framework for Inference Learning](https://mlsanity.com/paper/2206.00164/) (2022)
- [A Theoretical Framework for Inference and Learning in Predictive Coding Networks](https://mlsanity.com/paper/2207.12316/) (2022)
- [Predictive Coding beyond Gaussian Distributions](https://mlsanity.com/paper/2211.03481/) (2022)
- [Latent Equilibrium: A unified learning theory for arbitrarily fast computation with arbitrarily slow neurons](https://mlsanity.com/paper/2110.14549/) (2021)
- [Brain-Inspired Machine Intelligence: A Survey of Neurobiologically-Plausible Credit Assignment](https://mlsanity.com/paper/2312.09257/) (2023)
- [Predictive Coding Can Do Exact Backpropagation on Convolutional and Recurrent Neural Networks](https://mlsanity.com/paper/2103.03725/) (2021)
- [BayesPCN: A Continually Learnable Predictive Coding Associative Memory](https://mlsanity.com/paper/2205.09930/) (2022)
- [How to Train Your Wide Neural Network Without Backprop: An Input-Weight Alignment Perspective](https://mlsanity.com/paper/2106.08453/) (2021)
- [Predictive coding feedback results in perceived illusory contours in a recurrent neural network](https://mlsanity.com/paper/2102.01955/) (2021)
- [Variational Predictive Routing with Nested Subjective Timescales](https://mlsanity.com/paper/2110.11236/) (2021)
- [Is Bio-Inspired Learning Better than Backprop? Benchmarking Bio Learning vs. Backprop](https://mlsanity.com/paper/2212.04614/) (2022)
- [Predictive Coding as a Neuromorphic Alternative to Backpropagation: A Critical Evaluation](https://mlsanity.com/paper/2304.02658/) (2023)
- [A General Framework for Interpretable Neural Learning based on Local Information-Theoretic Goal Functions](https://mlsanity.com/paper/2306.02149/) (2023)
- [Spiking Neural Networks and Bio-Inspired Supervised Deep Learning: A Survey](https://mlsanity.com/paper/2307.16235/) (2023)
- [Predictive Coding Networks and Inference Learning: Tutorial and Survey](https://mlsanity.com/paper/2407.04117/) (2024)
- [Fast On-Device Adaptation for Spiking Neural Networks via Online-Within-Online Meta-Learning](https://mlsanity.com/paper/2103.03901/) (2021)
- [Bayesian brains and the R'enyi divergence](https://mlsanity.com/paper/2107.05438/) (2021)

## Abstract (arXiv)

Backpropagation of error (backprop) is a powerful algorithm for training machine learning architectures through end-to-end differentiation. However, backprop is often criticised for lacking biological plausibility. Recently, it has been shown that backprop in multilayer-perceptrons (MLPs) can be approximated using predictive coding, a biologically-plausible process theory of cortical computation which relies only on local and Hebbian updates. The power of backprop, however, lies not in its instantiation in MLPs, but rather in the concept of automatic differentiation which allows for the optimisation of any differentiable program expressed as a computation graph. Here, we demonstrate that predictive coding converges asymptotically (and in practice rapidly) to exact backprop gradients on arbitrary computation graphs using only local learning rules. We apply this result to develop a straightforward strategy to translate core machine learning architectures into their predictive coding equivalents. We construct predictive coding CNNs, RNNs, and the more complex LSTMs, which include a non-layer-like branching internal graph structure and multiplicative interactions. Our models perform equivalently to backprop on challenging machine learning benchmarks, while utilising only local and (mostly) Hebbian plasticity. Our method raises the potential that standard machine learning algorithms could in principle be directly implemented in neural circuitry, and may also contribute to the development of completely distributed neuromorphic architectures.