# Node Perturbation Can Effectively Train Multi-Layer Neural Networks

arXiv: https://arxiv.org/abs/2310.00965 · HTML: https://mlsanity.com/paper/2310.00965/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 35 references that are in the map.
- First posted on arXiv 2 Oct 2023.

## Builds on (most cited first, 12 of 12)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [The Forward-Forward Algorithm: Some Preliminary Investigations](https://mlsanity.com/paper/2212.13345/) (2022)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Direct Feedback Alignment Provides Learning in Deep Neural Networks](https://mlsanity.com/paper/1609.01596/) (2016)
- [Gradients without Backpropagation](https://mlsanity.com/paper/2202.08587/) (2022)
- [Scaling Forward Gradient With Local Losses](https://mlsanity.com/paper/2210.03310/) (2022)
- [Error-driven Input Modulation: Solving the Credit Assignment Problem without a Backward Pass](https://mlsanity.com/paper/2201.11665/) (2022)
- [Natural Neural Networks](https://mlsanity.com/paper/1507.00210/) (2015)
- [Brain-Inspired Learning on Neuromorphic Substrates](https://mlsanity.com/paper/2010.11931/) (2020)
- [Error Forward-Propagation: Reusing Feedforward Connections to Propagate Errors in Deep Learning](https://mlsanity.com/paper/1808.03357/) (2018)
- [Whitening and second order optimization both make information in the dataset unusable during training, and can reduce or prevent generalization](https://mlsanity.com/paper/2008.07545/) (2020)
- [Constrained Parameter Inference as a Principle for Learning](https://mlsanity.com/paper/2203.13203/) (2022)

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

- [Conditioned Direct Feedback Alignment via Activity and Error Geometry](https://mlsanity.com/paper/2607.18574/) (2026)
- [A Unified Perspective on Optimization in Machine Learning and Neuroscience: From Gradient Descent to Neural Adaptation](https://mlsanity.com/paper/2510.18812/) (2025)

## Abstract (arXiv)

Backpropagation (BP) remains the dominant and most successful method for training parameters of deep neural network models. However, BP relies on two computationally distinct phases, does not provide a satisfactory explanation of biological learning, and can be challenging to apply for training of networks with discontinuities or noisy node dynamics. By comparison, node perturbation (NP), also known as activity-perturbed forward gradients, proposes learning by the injection of noise into network activations, and subsequent measurement of the induced loss change. NP relies on two forward (inference) passes, does not make use of network derivatives, and has been proposed as a model for learning in biological systems. However, standard NP is highly data inefficient and can be unstable due to its unguided noise-based search process. In this work, we develop a modern perspective on NP by relating it to the directional derivative and incorporating input decorrelation. We find that a closer alignment with directional derivatives together with input decorrelation at every layer theoretically and practically enhances performance of NP learning with large improvements in parameter convergence and much higher performance on the test data, approaching that of BP. Furthermore, our novel formulation allows for application to noisy systems in which the noise process itself is inaccessible, which is of particular interest for on-chip learning in neuromorphic systems.