We introduce PC-ALM, a local alternative to backpropagation. PC-ALM trains residual MLPs up to 1000 layers, nearly matching backprop's performance despite using only layer-local dynamics. PC-ALM equips each layer with a feedback control dynamical system that distributes and propagates supervision credit throughout a network.

Standard deep learning relies on backpropagation. The brain, however, cannot implement backpropagation exactly. One reason is "phase locking": backprop runs in three strict phases—forward, backward, weight update—requiring neurons to hold activations and wait for error signals. The brain has no known mechanism for such strict timing coordination.

PC-ALM replaces forward and backward passes with layer-local dynamical systems. Each layer couples only to its neighbors. When run to convergence, the dynamics distribute supervision credit quickly and accurately. PC-ALM extends standard predictive coding (PC) by introducing dual neurons (Lagrange multipliers) per layer, making each layer's local recurrence a PI feedback controller. In linear networks, the dual neurons converge to exact backprop credit signals despite using only local computation.

We compare PC-ALM to PC and backprop. PC-ALM successfully propagates supervision credit in 1000-layer networks, overcoming PC's signal decay while remaining layer-local. This may inform energy-efficient deep learning on neuromorphic hardware.