A scalable Walsh-Hadamard regularizer to overcome the low-degree spectral bias of neural networks

Ali Gorji, Andisheh Amrollahi, Andreas Krause

Keywords: neural network theory, neural network regularization, machine learning fundamental, spectral bias of neural networks, simplicity bias of neural networks

TL;DR: We empirically investigate the spectral bias of neural networks in learning only low-degree interactions and introduce a regularizer to remedy this.

Abstract:

Despite the capacity of neural nets to learn arbitrary functions, models trained through gradient descent often exhibit a bias towards ``simpler'' functions. Various notions of simplicity have been introduced to characterize this behavior. Here, we focus on the case of neural networks with discrete (zero-one), high-dimensional, inputs through the lens of their Fourier (Walsh-Hadamard) transforms, where the notion of simplicity can be captured through the degree of the Fourier coefficients. We empirically show that neural networks have a tendency to learn lower-degree frequencies.
We show how this spectral bias towards low-degree frequencies can in fact hurt the neural network's generalization on real-world datasets. To remedy this we propose a new scalable functional regularization scheme that aids the neural network to learn higher degree frequencies. Our regularizer also helps avoid erroneous identification of low-degree frequencies, which further improves generalization. We extensively evaluate our regularizer on synthetic datasets to gain insights into its behavior. Finally, we show significantly improved generalization on four different datasets compared to standard neural networks and other relevant baselines.