On the Spectral Bias of Neural Networks

Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, Aaron Courville

Abstract: Neural networks are known to be a class of highly expressive functions able to fit even random input-output mappings with 100% accuracy. In this work we present properties of neural networks that complement this aspect of expressivity. By using tools from Fourier analysis, we show that deep ReLU networks are biased towards low frequency functions, meaning that they cannot have local fluctuations without affecting their global behavior. Intuitively, this property is in line with the observation that over-parameterized networks find simple patterns that generalize across data samples. We also investigate how the shape of the data manifold affects expressivity by showing evidence that learning high frequencies gets easier with increasing manifold complexity, and present a theoretical understanding of this behavior. Finally, we study the robustness of the frequency components with respect to parameter perturbation, to develop the intuition that the parameters must be finely tuned to express high frequency functions.

Keywords: deep learning theory, fourier analysis

TL;DR: We investigate ReLU networks in the Fourier domain and demonstrate peculiar behaviour.

Code: [ nasimrahaman/SpectralBias](https://github.com/nasimrahaman/SpectralBias) + [ 1 community implementation](https://paperswithcode.com/paper/?openreview=r1gR2sC9FX)

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/on-the-spectral-bias-of-neural-networks/code)