How Implicit Regularization of ReLU Neural Networks Characterizes the Learned Function -- Part I: the 1-D Case of Two Layers with Random First Layer
This provides theoretical insights into implicit regularization for neural networks, which is foundational for ML/AI researchers, though incremental as it builds on existing regularization theory.
The paper tackles the problem of understanding implicit regularization in shallow ReLU neural networks with random first-layer weights, showing mathematically that L2-regularized regression corresponds to regularizing the second derivative in function space and that the trained network converges to smooth spline interpolation as hidden nodes increase.
In this paper, we consider one dimensional (shallow) ReLU neural networks in which weights are chosen randomly and only the terminal layer is trained. First, we mathematically show that for such networks L2-regularized regression corresponds in function space to regularizing the estimate's second derivative for fairly general loss functionals. For least squares regression, we show that the trained network converges to the smooth spline interpolation of the training data as the number of hidden nodes tends to infinity. Moreover, we derive a novel correspondence between the early stopped gradient descent (without any explicit regularization of the weights) and the smoothing spline regression.