# Understanding Predictive Coding as an Adaptive Trust-Region Method

arXiv: https://arxiv.org/abs/2305.18188 · HTML: https://mlsanity.com/paper/2305.18188/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 31 references that are in the map.
- First posted on arXiv 29 May 2023.

## Builds on (most cited first, 21 of 21)

- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [The Forward-Forward Algorithm: Some Preliminary Investigations](https://mlsanity.com/paper/2212.13345/) (2022)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Predictive Coding Approximates Backprop along Arbitrary Computation Graphs](https://mlsanity.com/paper/2006.04182/) (2020)
- [Predictive Coding: a Theoretical and Experimental Review](https://mlsanity.com/paper/2107.12979/) (2021)
- [Gradient Descent Can Take Exponential Time to Escape Saddle Points](https://mlsanity.com/paper/1705.10412/) (2017)
- [Error-driven Input Modulation: Solving the Credit Assignment Problem without a Backward Pass](https://mlsanity.com/paper/2201.11665/) (2022)
- [A Theoretical Framework for Target Propagation](https://mlsanity.com/paper/2006.14331/) (2020)
- [Predictive Coding: Towards a Future of Deep Learning beyond Backpropagation?](https://mlsanity.com/paper/2202.09467/) (2022)
- [Biologically Motivated Algorithms for Propagating Local Target Representations](https://mlsanity.com/paper/1805.11703/) (2018)
- [The Power of Normalization: Faster Evasion of Saddle Points](https://mlsanity.com/paper/1611.04831/) (2016)
- [Revisiting Normalized Gradient Descent: Fast Evasion of Saddle Points](https://mlsanity.com/paper/1711.05224/) (2017)
- [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)
- [Efficient approaches for escaping higher order saddle points in non-convex optimization](https://mlsanity.com/paper/1602.05908/) (2016)
- [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 Can Do Exact Backpropagation on Convolutional and Recurrent Neural Networks](https://mlsanity.com/paper/2103.03725/) (2021)

## Built on by (most cited first, 1 of 1)

- [Brain-inspired Computational Intelligence via Predictive Coding](https://mlsanity.com/paper/2308.07870/) (2023)

## Abstract (arXiv)

Predictive coding (PC) is a brain-inspired local learning algorithm that has recently been suggested to provide advantages over backpropagation (BP) in biologically relevant scenarios. While theoretical work has mainly focused on showing how PC can approximate BP in various limits, the putative benefits of "natural" PC are less understood. Here we develop a theory of PC as an adaptive trust-region (TR) algorithm that uses second-order information. We show that the learning dynamics of PC can be interpreted as interpolating between BP's loss gradient direction and a TR direction found by the PC inference dynamics. Our theory suggests that PC should escape saddle points faster than BP, a prediction which we prove in a shallow linear model and support with experiments on deeper networks. This work lays a foundation for understanding PC in deep and wide networks.