Expressivity of Neural Networks with Random Weights and Learned Biases

Abstract

Landmark universal function approximation results for neural networks with trained weights and biases provided impetus for the ubiquitous use of neural networks as learning models in Artificial Intelligence (AI) and neuroscience. Recent work has pushed the bounds of universal approximation by showing that arbitrary functions can similarly be learned by tuning smaller subsets of parameters, for example the output weights, within randomly initialized networks. Motivated by the fact that biases can be interpreted as biologically plausible mechanisms for adjusting unit outputs in neural networks, such as tonic inputs or activation thresholds, we investigate the expressivity of neural networks with random weights where only biases are optimized. We provide theoretical and numerical evidence demonstrating that feedforward neural networks with fixed random weights can be trained to perform multiple tasks by learning biases only. We further show that an equivalent result holds for recurrent neural networks predicting dynamical system trajectories. Our results are relevant to neuroscience, where they demonstrate the potential for behaviourally relevant changes in dynamics without modifying synaptic weights, as well as for AI, where they shed light on multi-task methods such as bias fine-tuning and unit masking.