On the Activation Function Dependence of the Spectral Bias of Neural Networks

Qingguo Hong, Jonathan W. Siegel, Qinyang Tan, Jinchao Xu

Abstract: Neural networks are universal function approximators which are known to generalize well despite being dramatically overparameterized. We study this phenomenon from the point of view of the spectral bias of neural networks. Our contributions are two-fold. First, we provide a theoretical explanation for the spectral bias of ReLU neural networks by leveraging connections with the theory of finite element methods, which is widely used to numerically solve PDEs. Second, based upon this theory we predict that switching the activation function to a piecewise linear B-spline, namely the Hat function, will remove this spectral bias, which we verify empirically in a variety of settings. This is of particular significance for solving PDEs using neural networks since for such problems it is important to capture all frequencies in the solutions. Our empirical studies also show that neural networks with the Hat activation function are trained significantly faster using stochastic gradient descent and ADAM. Combined with previous work showing that the Hat activation function also improves generalization accuracy on image classification tasks, this indicates that using the Hat activation provides significant advantages over the ReLU on a variety of problems.

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