ePC: Overcoming Exponential Signal Decay in Deep Predictive Coding Networks

Predictive Coding (PC) offers a biologically plausible alternative to backpropagation for neural network training, yet struggles with deeper architectures. This paper identifies the root cause and provides a principled solution. We uncover that the canonical state-based formulation of PC (sPC) is, by design, deeply inefficient on digital hardware, due to an inherent signal decay problem that scales exponentially with depth. To address this fundamental limitation, we introduce a novel reparameterization of PC, named error-based PC (ePC), which does not suffer from signal decay. By optimizing over prediction errors rather than states, ePC enables signals to reach all layers simultaneously and unattenuated, converging orders of magnitude faster than sPC. Experiments across multiple architectures and datasets demonstrate that ePC matches backpropagation's performance even for deeper models where sPC struggles. Besides practical improvements, our work provides theoretical insight into PC dynamics and establishes a foundation for scaling bio-inspired learning to deeper architectures on digital hardware and beyond.