# Closed-Form Feedback-Free Learning with Forward Projection

arXiv: https://arxiv.org/abs/2501.16476 · HTML: https://mlsanity.com/paper/2501.16476/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 53 references that are in the map.
- First posted on arXiv 27 Jan 2025.

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [The Forward-Forward Algorithm: Some Preliminary Investigations](https://mlsanity.com/paper/2212.13345/) (2022)
- [Difference Target Propagation](https://mlsanity.com/paper/1412.7525/) (2014)
- [Training Neural Networks with Local Error Signals](https://mlsanity.com/paper/1901.06656/) (2019)
- [Explainable AI (XAI): A Systematic Meta-Survey of Current Challenges and Future Opportunities](https://mlsanity.com/paper/2111.06420/) (2021)
- [On the Power and Limitations of Random Features for Understanding Neural Networks](https://mlsanity.com/paper/1904.00687/) (2019)
- [Deep Learning without Weight Transport](https://mlsanity.com/paper/1904.05391/) (2019)
- [Predictive Coding: a Theoretical and Experimental Review](https://mlsanity.com/paper/2107.12979/) (2021)
- [Revisiting Locally Supervised Learning: an Alternative to End-to-end Training](https://mlsanity.com/paper/2101.10832/) (2021)
- [Error-driven Input Modulation: Solving the Credit Assignment Problem without a Backward Pass](https://mlsanity.com/paper/2201.11665/) (2022)
- [Biologically Motivated Algorithms for Propagating Local Target Representations](https://mlsanity.com/paper/1805.11703/) (2018)
- [SpikingResformer: Bridging ResNet and Vision Transformer in Spiking Neural Networks](https://mlsanity.com/paper/2403.14302/) (2024)
- [Deep learning with asymmetric connections and Hebbian updates](https://mlsanity.com/paper/1812.07965/) (2018)
- [Forward Learning with Top-Down Feedback: Empirical and Analytical Characterization](https://mlsanity.com/paper/2302.05440/) (2023)
- [Biologically Plausible Training Mechanisms for Self-Supervised Learning in Deep Networks](https://mlsanity.com/paper/2109.15089/) (2021)
- [LLS: Local Learning Rule for Deep Neural Networks Inspired by Neural Activity Synchronization](https://mlsanity.com/paper/2405.15868/) (2024)
- [Baseline Drift Tolerant Signal Encoding for ECG Classification with Deep Learning](https://mlsanity.com/paper/2405.00724/) (2024)

## Built on by (most cited first, 2 of 2)

- [SAL: Selective Adaptive Learning for Backpropagation-Free Training with Sparsification](https://mlsanity.com/paper/2601.21561/) (2026)
- [The Price of Locality: Why Forward-Forward Underperforms Backpropagation?](https://mlsanity.com/paper/2609.33240/) (2026)

## Abstract (arXiv)

State-of-the-art backpropagation-free learning methods employ local error feedback to direct iterative optimisation via gradient descent. Here, we examine the more restrictive setting where retrograde communication from neuronal outputs is unavailable for pre-synaptic weight optimisation. We propose Forward Projection (FP), a randomised closed-form training method requiring only a single forward pass over the dataset without retrograde communication. FP generates target values for pre-activation membrane potentials through randomised nonlinear projections of pre-synaptic inputs and labels. Local loss functions are optimised using closed-form regression without feedback from downstream layers. A key advantage is interpretability: membrane potentials in FP-trained networks encode information interpretable layer-wise as label predictions. Across several biomedical datasets, FP achieves generalisation comparable to gradient descent-based local learning methods while requiring only a single forward propagation step, yielding significant training speedup. In few-shot learning tasks, FP produces more generalisable models than backpropagation-optimised alternatives, with local interpretation functions successfully identifying clinically salient diagnostic features.