Closed-Form Feedback-Free Learning with Forward Projection (2501.16476v1)

Abstract: State-of-the-art methods for backpropagation-free learning employ local error feedback to direct iterative optimisation via gradient descent. In this study, we examine the more restrictive setting where retrograde communication from neuronal outputs is unavailable for pre-synaptic weight optimisation. To address this challenge, we propose Forward Projection (FP). This novel randomised closed-form training method requires only a single forward pass over the entire dataset for model fitting, without retrograde communication. Target values for pre-activation membrane potentials are generated layer-wise via nonlinear projections of pre-synaptic inputs and the labels. Local loss functions are optimised over pre-synaptic inputs using closed-form regression, without feedback from neuronal outputs or downstream layers. Interpretability is a key advantage of FP training; membrane potentials of hidden neurons in FP-trained networks encode information which is interpretable layer-wise as label predictions. We demonstrate the effectiveness of FP across four biomedical datasets. In few-shot learning tasks, FP yielded more generalisable models than those optimised via backpropagation. In large-sample tasks, FP-based models achieve generalisation comparable to gradient descent-based local learning methods while requiring only a single forward propagation step, achieving significant speed up for training. Interpretation functions defined on local neuronal activity in FP-based models successfully identified clinically salient features for diagnosis in two biomedical datasets. Forward Projection is a computationally efficient machine learning approach that yields interpretable neural network models without retrograde communication of neuronal activity during training.