Sakana AI researchers introduced PC-ALM (Augmented Lagrangian Predictive Coding), a layer‑local training method that augments standard predictive coding with one Lagrange multiplier per layer. This approach preserves the locality of updates while allowing the multipliers to accumulate into credit signals aligned with backpropagation, enabling effective training of very deep, narrow networks such as 1000‑layer models. The technique bridges the gap between biologically plausible local learning and the performance of conventional backprop.