Closed-form predictive coding via hierarchical Gaussian filters
Authors: Aleksandrs Baskakovs, Sylvain Estebe, Kenneth Enevoldsen et al.
Summary
arXiv:2605. 20293v1 Announce Type: new 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.
Relevance
Read next because Closed-form predictive coding via hierarchical Gaussian filters overlaps with clean result "Language-mismatch LoRA SFT on Qwen2.5-7B leaks the trained completion language into bystander directives the model was never trained on, absent under same-language SFT (LOW confidence)", clean result "Training one persona to emit a [ZLT] marker without bystanders adopting it has a one-cell-wide LR x epochs window on Qwen2.5-7B-Instruct (LOW confidence)", clean result "The marker is a representational handle, not a behavioural one — sharing it between a villain persona and the assistant transfers no misalignment (HIGH confidence)". Matching terms: under, epochs, line, without. Source: arxiv cs.LG (Machine Learning).
Abstract
arXiv:2605.20293v1 Announce Type: new 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.