Accelerated Predictive Coding Networks via Direct Kolen-Pollack Feedback Alignment

Predictive coding (PC) is a biologically inspired algorithm for training neural networks that relies only on local updates, allowing parallel learning across layers. However, practical implementations face two key limitations: error signals must still propagate from the output to early layers through multiple inference-phase steps, and feedback decays exponentially during this process, leading to vanishing updates in early layers. We propose direct Kolen-Pollack predictive coding (DKP-PC), which simultaneously addresses both feedback delay and exponential decay, yielding a more efficient and scalable variant of PC while preserving update locality. Leveraging direct feedback alignment and direct Kolen-Pollack algorithms, DKP-PC introduces learnable feedback connections from the output layer to all hidden layers, establishing a direct pathway for error transmission. This yields an algorithm that reduces the theoretical error propagation time complexity from O(L), with L being the network depth, to O(1), removing depth-dependent delay in error signals. Moreover, empirical results demonstrate that DKP-PC achieves performance at least comparable to, and often exceeding, that of standard PC, while offering improved latency and computational performance, supporting its potential for custom hardware-efficient implementations.

Bio-Inspired Artificial Neural Networks based on Predictive Coding

Backpropagation (BP) of errors is the backbone training algorithm for artificial neural networks (ANNs). It updates network weights through gradient descent to minimize a loss function representing the mismatch between predictions and desired outputs. BP uses the chain rule to propagate the loss gradient backward through the network hierarchy, allowing efficient weight updates. However, this process requires weight updates at every layer to rely on a global error signal generated at the network's output. In contrast, the Hebbian model of synaptic plasticity states that weight updates are local, depending only on the activity of pre- and post-synaptic neurons. This suggests biological brains likely do not implement BP directly. Recently, Predictive Coding (PC) has gained interest as a biologically plausible alternative that updates weights using only local information. Originating from 1950s work on signal compression, PC was later proposed as a model of the visual cortex and formalized under the free energy principle, linking it to Bayesian inference and dynamical systems. PC weight updates rely solely on local information and provide theoretical advantages such as automatic scaling of gradients based on uncertainty. This lecture notes column offers a novel, tutorial-style introduction to PC, focusing on its formulation, derivation, and connections to well-known optimization and signal processing algorithms such as BP and the Kalman Filter (KF). It aims to support existing literature by guiding readers from the mathematical foundations of PC to practical implementation, including Python examples using PyTorch.