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A Study of Hidden-State Optimization Order in Predictive Coding Networks
Xueyuan Li, Danilo Vasconcellos Vargas
TL;DR
Deep local-learning systems can suffer from weak feature learning when local objectives provide limited or imbalanced updates. This paper tests whether changing hidden-state optimization order helps by coordinating chunk boundaries before refining within chunks, and reports improved CIFAR-10 accuracy together with stronger feature-learning diagnostics.
Problem
Local objectives in deep local-learning systems can produce weak or imbalanced updates, motivating study of how objective organization affects feature learning.
Method
The paper partitions networks into chunks, coordinates boundary states through inter-chunk inference, and then refines within-chunk activities in predictive-coding networks.
Results
9.77% and 5.51%: boundary-first predictive coding improves CIFAR-10 accuracy over standard predictive coding under standard and μ-parametrizations, respectively.
Takeaways & Limitations
Boundary-first, chunk-based inference is supported as a useful schedule for predictive-coding training, with more active early-layer updates, lower CKA, and more diverse layerwise gradients.
Takeaways & Limitations
The optimization schedule is studied in predictive-coding networks, while extension to other local-learning frameworks remains future work.
Abstract
from arXiv · showhide
Local learning methods offer an alternative to end-to-end backpropagation, but their unstructured local objectives can produce weak feature learning in deep networks. We study whether the order of hidden-state optimization can address this limitation. We propose a boundary-first inference schedule that partitions a model into chunks, first coordinates hidden states at chunk boundaries, and then refines representations within each chunk. We instantiate this schedule in predictive coding networks (PCNs), a local-learning framework in which hidden activities and prediction errors are explicitly exposed during inference. On CIFAR-10, the resulting boundary-first predictive-coding instantiation improves accuracy over standard predictive coding by $9.77\%$ under a standard parametrization and by $5.51\%$ under a $μ$-parametrization. Diagnostic analyses further show more non-trivial early-layer updates, lower initial-to-final CKA, and more diverse layerwise gradients, consistent with stronger feature learning. These results support boundary-first, chunk-based inference as a practical design principle for predictive-coding training and motivate its study in broader local-learning systems.
1 Introduction
Local learning replaces global backpropagation with local objectives, but predictive-coding systems can produce weak updates in early and intermediate layers. This work changes hidden-state optimization order through boundary-first, chunk-based inference and reports improved updates, performance, and feature-learning diagnostics.
- Local learning offers modular and biologically plausible alternatives to end-to-end backpropagation, which propagates a global supervised objective through all layers.
- Weak or degenerate early-layer updates and unstable inference dynamics remain reported failure modes of deep predictive-coding networks.
- Boundary-first inference coordinates hidden states at chunk boundaries before refining remaining activities within each chunk.
- The schedule is instantiated in predictive coding as PC-PC and evaluated with PC-BP, which uses predictive-coding boundary coordination but backpropagation within chunks.
- 9.77% and 5.51%: the boundary-first instantiation improves standard predictive coding on CIFAR-10 under standard and μ-parametrizations, respectively.The standard parametrization comparison is reported for standard PC, while the μ-parametrization comparison is reported for the μMLP setting.
- All 20 layers receive non-trivial updates, while standard predictive coding updates only 4/20 MLP layers and 5/20 μMLP layers.
- PC-PC and PC-BP reduce initial-to-final CKA relative to standard predictive coding and produce lower gradient similarity than global backpropagation in measured settings.
2 Related Work
Related work frames predictive coding as local-error learning, chunking as a way to organize variable groups, and CKA and gradient similarity as diagnostics of representation change and update diversity.
- Predictive-coding networks minimize prediction errors between adjacent layers and have been developed as a broader local-learning framework.
- Chunking groups lower-level elements into higher-level units and motivates blockwise, modular, and greedy layerwise training strategies.
- Feature learning is distinguished from kernel-like learning by whether representations change during training rather than remaining nearly fixed.
- CKA measures similarity between initial and final representations, with lower CKA indicating greater representation change.
- Layerwise update diversity is measured by cosine similarity between weight gradients, with lower similarity indicating more diverse updates.
3 Methodology
The method changes predictive-coding inference by coordinating chunk-boundary states before refining interior activities, while retaining a hybrid backpropagation control.
- Standard predictive coding: Predictive coding minimizes local prediction errors together with a supervised output loss while inferring hidden activities.Hidden activities are updated with weights fixed, using neighboring layers through local error terms.
- Boundary-first schedule: The boundary-first schedule partitions the network into consecutive chunks containing boundary and non-boundary variables.It first coordinates boundary states between chunks, then refines hidden activities within each chunk.
- Boundary-first schedule: Inter-chunk inference aligns boundary states by minimizing prediction errors for boundary variables together with the supervised output loss.Chunks are asked to agree on their boundaries before internal activities are refined.
- Boundary-first schedule: Intra-chunk predictive-coding inference refines non-boundary activities using prediction errors from adjacent layers within each chunk.The weights are then updated using the inferred activities; this fully local instantiation is called PC-PC.
- Hybrid control: PC-BP preserves boundary alignment but updates weights inside each chunk with backpropagation as a hybrid control.Its purpose is to test whether the boundary-first schedule remains useful without purely predictive-coding learning inside chunks.
4 Experiments
Experiments compare global backpropagation, standard predictive coding, PC-PC, and PC-BP on MNIST and CIFAR-10 using MLP parameterizations and diagnostic feature-learning measures.
- Experimental setup: Experiments use standard MLPs and µMLPs trained with four dynamics: global backpropagation, standard predictive coding, PC-PC, and PC-BP.The evaluations are conducted on MNIST and CIFAR-10.
- Experimental setup: The main experiments use 20-layer MLPs with 500 hidden units per layer, divided into four five-layer chunks for PC-PC and PC-BP.Predictive-coding methods use 30 inference steps; PC-PC allocates 15 steps to each inference stage.
- Evaluation: The study measures test accuracy, initial-to-final representation CKA, and between-layer gradient similarity.Depth and chunk size are also varied to test dependence on a particular partition.
5 Results
Across MNIST and CIFAR-10 experiments, boundary-first predictive coding improved performance and changed representation and update dynamics relative to standard predictive coding. The schedule was most beneficial across most tested depths and chunk sizes, except in the 5-layer setting.
- Performance: On CIFAR-10, PC-PC improved standard MLP accuracy from 41.67% to 51.44%, a 9.77% gain, while PC-BP improved µMLP accuracy from 48.77% to 54.28%, a 5.51% gain.The gains on MNIST were small, consistent with the task’s linear separability.
- Representation change: PC-PC and PC-BP reduced initial-to-final CKA on MNIST and CIFAR-10, indicating greater representation change than standard predictive coding.Lower CKA was used as the measure of greater representation change.
- Gradient alignment: All predictive-coding variants had lower gradient alignment with global backpropagation than 0.7, so boundary-first methods did not simply reproduce global BP updates.The µMLP showed higher alignment than the standard MLP, and PC-BP showed higher alignment than PC-PC.
- Layerwise updates: Boundary-first PCs activated early layers and reduced similarity between layerwise updates, whereas standard PC left several early layers with weak or near-zero gradients.Across measured settings, predictive-coding dynamics had lower gradient similarity than global backpropagation, indicating more distinct update directions.
- Depth and chunk size: Across most measured CIFAR-10 depths and chunk sizes, boundary-first inference improved over standard predictive coding, except in the 5-layer setting.Chunk size 1 represents standard PC, while a chunk size equal to network depth represents global backpropagation.
6 Conclusion
The study evaluates boundary-first, chunk-based hidden-state optimization in predictive-coding networks and finds empirical improvements over standard predictive coding. The evidence remains specific to the studied PCN instantiations, with broader frameworks, architectures, and tasks left for future work.
- Contribution: The boundary-first schedule aligns chunk-boundary hidden states before refining activities within each chunk.In predictive coding, this schedule produced increased early-layer update activity, lower initial-to-final CKA, and more diverse layerwise gradients.
- Findings: Across MNIST and CIFAR-10, boundary-first inference improved standard predictive coding and altered representation and gradient dynamics.The reported changes include increased early-layer updates, reduced initial-to-final CKA, and more diverse layerwise gradients.
- Scope: The evidence is empirical and specific to the predictive-coding instantiations studied in this work.Extension to other local-learning frameworks, architectures, and tasks remains future work.
A Standard Deviations of Accuracies in Table 1
Table 2 reports the standard deviations of the test accuracies presented in Table 1.
- Reported variability: Table 2 contains standard deviations for the test accuracies reported in Table 1.It supplements the accuracy values for standard predictive coding, PC-PC, and PC-BP on MNIST and CIFAR-10.
B Feature Diversity under Boundary-First Predictive Coding on CIFAR-10
On CIFAR-10, boundary-first predictive coding produces more broadly distributed and more diverse layerwise updates than standard predictive coding.
- Standard predictive coding leaves several early layers with weak gradients during CIFAR-10 training.
- Boundary-first predictive coding activates more layers and reduces similarity among their updates.
- Lower gradient similarity indicates more diverse layerwise updates in the CIFAR-10 analysis.