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In this paper, we consider one dimensional (shallow) ReLU neural networks inwhich weights are chosen randomly and only the terminal layer is trained.First, we mathematically show that for such networks L2-regularized regressioncorresponds in function space to regularizing the estimate's second derivativefor fairly general loss functionals. For least squares regression, we show thatthe trained network converges to the smooth spline interpolation of thetraining data as the number of hidden nodes tends to infinity. Moreover, wederive a novel correspondence between the early stopped gradient descent(without any explicit regularization of the weights) and the smoothing splineregression.

How Implicit Regularization of ReLU Neural Networks Characterizes the Learned Function -- Part I: the 1-D Case of Two Layers with Random First Layer