El Mehdi Achour

Francesco Innocenti, El Mehdi Achour, Ryan Singh et al.

Predictive coding (PC) is an energy-based learning algorithm that performs iterative inference over network activities before updating weights. Recent work suggests that PC can converge in fewer learning steps than backpropagation thanks to its inference procedure. However, these advantages are not always observed, and the impact of PC inference on learning is not theoretically well understood. Here, we study the geometry of the PC energy landscape at the inference equilibrium of the network activities. For deep linear networks, we first show that the equilibrated energy is simply a rescaled mean squared error loss with a weight-dependent rescaling. We then prove that many highly degenerate (non-strict) saddles of the loss including the origin become much easier to escape (strict) in the equilibrated energy. Our theory is validated by experiments on both linear and non-linear networks. Based on these and other results, we conjecture that all the saddles of the equilibrated energy are strict. Overall, this work suggests that PC inference makes the loss landscape more benign and robust to vanishing gradients, while also highlighting the fundamental challenge of scaling PC to deeper models.

Francesco Innocenti, El Mehdi Achour, Christopher L. Buckley

The biological implausibility of backpropagation (BP) has motivated many alternative, brain-inspired algorithms that attempt to rely only on local information, such as predictive coding (PC) and equilibrium propagation. However, these algorithms have notoriously struggled to train very deep networks, preventing them from competing with BP in large-scale settings. Indeed, scaling PC networks (PCNs) has recently been posed as a challenge for the community (Pinchetti et al., 2024). Here, we show that 100+ layer PCNs can be trained reliably using a Depth-$μ$P parameterisation (Yang et al., 2023; Bordelon et al., 2023) which we call "$μ$PC". Through an extensive analysis of the scaling behaviour of PCNs, we reveal several pathologies that make standard PCNs difficult to train at large depths. We then show that, despite addressing only some of these instabilities, $μ$PC allows stable training of very deep (up to 128-layer) residual networks on simple classification tasks with competitive performance and little tuning compared to current benchmarks. Moreover, $μ$PC enables zero-shot transfer of both weight and activity learning rates across widths and depths. Our results have implications for other local algorithms and could be extended to convolutional and transformer architectures. Code for $μ$PC is made available as part of a JAX library for PCNs at https://github.com/thebuckleylab/jpc (Innocenti et al., 2024).