Closed-form predictive coding via hierarchical Gaussian filters

TL;DR

This paper introduces a novel closed-form predictive coding framework using hierarchical Gaussian filters that improves training speed, efficiency, and biological plausibility in deep neural networks.

Contribution

It reformulates predictive coding networks as deep hierarchical Gaussian filters, enabling fast, local, Bayesian inference with online learning of activations, weights, and precisions.

Findings

Approaches backpropagation in efficiency on FashionMNIST

Outperforms backpropagation on online and data efficiency tasks

Converges faster and handles concept drift effectively

Abstract

Predictive coding (PC) offers a local and biologically grounded alternative to backpropagation in the training of artificial neural networks, yet to date, it remains slower, and performance degrades sharply as network depth increases. We trace both problems to a single simplification: current PC networks fix the precision matrix to the identity, discarding precision-weighted prediction errors that the variational derivation requires to be fast, local, and Bayesian. We close this gap by expressing predictive coding networks as deep hierarchical Gaussian filters (HGFs) and restore precision-weighted message passing, yielding dynamic uncertainty estimates and Hebbian-compatible update rules at every layer. The resulting networks can simultaneously learn activations, weights, and precisions under a single free-energy objective, with no global error signal, and resolve inference without requiring iterations or automatic differentiation. On FashionMNIST, our solution approaches backpropagation in epoch-level wall-clock cost while converging in fewer epochs, and outperforms it on online, data efficiency, and concept-drift tasks. We thus establish that closed-form variational inference with online precision learning provides a tractable foundation for deep predictive coding networks, retaining biological and interpretative advantages, without requiring iterative relaxation or global error signals.