Function Forms of Simple ReLU Networks with Random Hidden Weights
This work advances the theoretical framework for understanding optimization and expressivity in wide neural networks, though it is incremental in refining existing concepts.
The paper tackles the function space dynamics of two-layer ReLU networks in the infinite-width limit, showing that basis functions converge to distinct forms prioritized by gradient descent, with simulations validating theoretical approximations.
We investigate the function space dynamics of a two-layer ReLU neural network in the infinite-width limit, highlighting the Fisher information matrix (FIM)'s role in steering learning. Extending seminal works on approximate eigendecomposition of the FIM, we derive the asymptotic behavior of basis functions ($f_v(x) = X^{\top} v $) for four groups of approximate eigenvectors, showing their convergence to distinct function forms. These functions, prioritized by gradient descent, exhibit FIM-induced inner products that approximate orthogonality in the function space, forging a novel connection between parameter and function spaces. Simulations validate the accuracy of these theoretical approximations, confirming their practical relevance. By refining the function space inner product's role, we advance the theoretical framework for ReLU networks, illuminating their optimization and expressivity. Overall, this work offers a robust foundation for understanding wide neural networks and enhances insights into scalable deep learning architectures, paving the way for improved design and analysis of neural networks.