This article introduces PC-ALM (Augmented Lagrangian Predictive Coding), a method designed to train residual multilayer perceptrons (MLPs) with up to 1000 layers, aiming to match the performance of traditional backpropagation while relying solely on layer-local dynamics. The method incorporates a feedback control dynamical system in each layer to efficiently distribute and propagate supervision credit throughout the network.
Traditional deep learning methods primarily utilize backpropagation, which is not directly implementable by the brain. The inability of the brain to execute exact backpropagation raises questions about how it addresses the multilayer credit assignment problem, a significant topic in neuroscience.
Backpropagation operates in three distinct phases: a forward pass, a backward pass, and a weight update. This strict timing coordination is not feasible in biological systems, which lack mechanisms to enforce such synchronization across networks.
PC-ALM replaces the forward and backward passes of backpropagation with local dynamical systems that couple each layer only to its immediate neighbors. Instead of executing a forward-then-backward sequence, PC-ALM runs each layer forward in time, allowing for rapid and accurate distribution of supervision credit signals across the entire network.
The method builds upon standard predictive coding (PC), which employs diffusive coupling between layers. PC-ALM enhances this by introducing dual neurons, or Lagrange multipliers, for each layer, transforming the local recurrence into a proportional-integral (PI) feedback controller. In linear networks, these dual neurons converge to the exact backpropagation credit signals while relying solely on local computations.
The authors conducted experiments comparing PC-ALM with traditional PC and backpropagation using simple tasks such as Fashion-MNIST and CIFAR-10. The focus was on deep, narrow networks where PC typically underperforms. The research aims to explore how distributed systems like the brain can perform gradient computations without backpropagation, with potential implications for energy-efficient deep learning on neuromorphic hardware.
PC-ALM is presented as a quadratic relaxation of the constrained optimization problem, where the network's depth, input, target, weights, and activations are defined. The training process alternates between inference and learning steps, with the weight update being Hebbian-like, reflecting a delta rule.
The results indicate that PC-ALM can successfully train 1000-layer MLPs on MNIST and improve performance over PC in various image classification tasks, including ResNet-18 on CIFAR-10 and Tiny ImageNet. Notably, PC-ALM demonstrates unique dynamical properties, with credit signals propagating through the network more rapidly than in traditional PC methods.
In conclusion, PC-ALM represents a significant advancement in training deep networks without backpropagation, providing insights into credit assignment mechanisms in biological systems and offering a promising avenue for future research in neural network training methodologies.