On the Infinite Width and Depth Limits of Predictive Coding Networks

Predictive coding (PC) is a biologically plausible alternative to standard backpropagation (BP) that minimises an energy function with respect to network activities before updating weights. Recent work has improved the training stability of deep PC networks (PCNs) by leveraging some BP-inspired reparameterisations, but the scalability and theoretical basis of these methods remain unclear. To address this gap, we study the infinite width and depth limits of PCNs. For linear networks, we derive stable and "non-lazy" parameterisations when scaling both the model width and depth, revealing that the output of standard PCNs explodes with width during training. Moreover, under stable parameterisations, we show that the gradients computed by PC at activity equilibrium converge to the BP gradients for networks that are much wider than deep ($depth/width\to0$). Experiments show high gradient alignment between PC and BP at large width for different nonlinear models, including convolutional networks and transformers. Overall, this work constrains the parameterisations that are scalable with PC, while suggesting how BP could be implemented using only local updates in much wider than deep networks like the brain.