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

Jakob Heiss; Josef Teichmann; Hanna Wutte

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arXiv · 1911.02903

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

Jakob Heiss·Josef Teichmann·Hanna Wutte

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Abstract

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.

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Jakob Heiss, Josef Teichmann, Hanna Wutte. 2019-11-07. How Implicit Regularization of ReLU Neural Networks Characterizes the Learned Function -- Part I: the 1-D Case of Two Layers with Random First Layer. https://arxiv.org/abs/1911.02903

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