# Augmented Lagrangian Predictive Coding

arXiv: https://arxiv.org/abs/2605.31022 · HTML: https://mlsanity.com/paper/2605.31022/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 46 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 29 of 29)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Difference Target Propagation](https://mlsanity.com/paper/1412.7525/) (2014)
- [Predictive Coding Approximates Backprop along Arbitrary Computation Graphs](https://mlsanity.com/paper/2006.04182/) (2020)
- [How Auto-Encoders Could Provide Credit Assignment in Deep Networks via Target Propagation](https://mlsanity.com/paper/1407.7906/) (2014)
- [Predictive Coding: a Theoretical and Experimental Review](https://mlsanity.com/paper/2107.12979/) (2021)
- [Training Neural Networks Without Gradients: A Scalable ADMM Approach](https://mlsanity.com/paper/1605.02026/) (2016)
- [Learning on Arbitrary Graph Topologies via Predictive Coding](https://mlsanity.com/paper/2201.13180/) (2022)
- [Benchmarking Predictive Coding Networks -- Made Simple](https://mlsanity.com/paper/2407.01163/) (2024)
- [Lifted Neural Networks](https://mlsanity.com/paper/1805.01532/) (2018)
- [ADMM for Efficient Deep Learning with Global Convergence](https://mlsanity.com/paper/1905.13611/) (2019)
- [On the relationship between predictive coding and backpropagation](https://mlsanity.com/paper/2106.13082/) (2021)
- [Fenchel Lifted Networks: A Lagrange Relaxation of Neural Network Training](https://mlsanity.com/paper/1811.08039/) (2018)
- [A Theoretical Framework for Inference and Learning in Predictive Coding Networks](https://mlsanity.com/paper/2207.12316/) (2022)
- [On ADMM in Deep Learning: Convergence and Saturation-Avoidance](https://mlsanity.com/paper/1902.02060/) (2019)
- [μPC: Scaling Predictive Coding to 100+ Layer Networks](https://mlsanity.com/paper/2505.13124/) (2025)
- [Lifted Proximal Operator Machines](https://mlsanity.com/paper/1811.01501/) (2018)
- [Dual Propagation: Accelerating Contrastive Hebbian Learning with Dyadic Neurons](https://mlsanity.com/paper/2302.01228/) (2023)
- [Understanding and Improving Optimization in Predictive Coding Networks](https://mlsanity.com/paper/2305.13562/) (2023)
- [Lifted Bregman Training of Neural Networks](https://mlsanity.com/paper/2208.08772/) (2022)
- [Only Strict Saddles in the Energy Landscape of Predictive Coding Networks?](https://mlsanity.com/paper/2408.11979/) (2024)
- [Contrastive Learning for Lifted Networks](https://mlsanity.com/paper/1905.02507/) (2019)
- [Neural Network Training as an Optimal Control Problem: An Augmented Lagrangian Approach](https://mlsanity.com/paper/2103.14343/) (2021)
- [Sheaf Cohomology of Linear Predictive Coding Networks](https://mlsanity.com/paper/2511.11092/) (2025)
- [A Physical Theory of Backpropagation: Exact Gradients from the Least-Action Principle](https://mlsanity.com/paper/2602.02281/) (2026)
- [On the Infinite Width and Depth Limits of Predictive Coding Networks](https://mlsanity.com/paper/2602.07697/) (2026)
- [Neural Networks as Local-to-Global Computations](https://mlsanity.com/paper/2603.14831/) (2026)
- [An Augmented Lagrangian Method for Training Recurrent Neural Networks](https://mlsanity.com/paper/2402.13687/) (2024)
- [A Unified Framework for Lifted Training and Inversion Approaches](https://mlsanity.com/paper/2510.09796/) (2025)

## Built on by (most cited first, 1 of 1)

- [Analog-Friendly Predictive Coding without Activation Derivatives](https://mlsanity.com/paper/2609.32350/) (2026)

## Abstract (arXiv)

Predictive coding (PC) is a local-learning alternative to backpropagation (BP), training deep networks via local energy-minimization dynamics rather than a global backward pass. We introduce Augmented Lagrangian Predictive Coding (PC-ALM), which maintains PC's inference budget but aligns each weight update toward BP by accumulating per-layer constraint errors into a layer-local Lagrange multiplier. In linear PC networks, PC-ALM converges to an equilibrium with exact BP gradients distributed across the network via only layer-local updates. We analyze PC-ALM in nonlinear PC networks up to depth 128 and show that it matches BP performance across all width-depth regimes, notably in deep narrow networks where PC underperforms. PC-ALM introduces recurrent dynamics in each layer's activations. Compared to PC's heat flow on a scalar energy, PC-ALM dynamics are driven by dual ascent on the augmented Lagrangian. We observe "ballistic" credit propagation across very deep networks, with credit signals evenly distributed across layers, compared to PC's slow, diffusive credit propagation. Beyond the algorithm itself, the augmented Lagrangian framework offers a generalization of PC, and may yield insights into how distributed systems could compute and propagate BP-like credit signals through purely local dynamics.