Towards Understanding the Spectral Bias of Deep Learning
Yuan Cao, Zhiying Fang, Yue Wu, Ding-Xuan Zhou, Quanquan Gu
TL;DR
The paper provides a rigorous NTK-based explanation for spectral bias, showing that gradient descent in over-parameterized two-layer ReLU networks effectively learns target components along NTK eigen-directions with rates governed by the corresponding eigenvalues. It establishes a general convergence theorem for arbitrary data and delivers a detailed spectral analysis for inputs uniformly distributed on the unit sphere, where low-degree spherical harmonics correspond to the high-eigenvalue directions and thus are learned faster. The authors corroborate the theory with experiments that demonstrate the expected learning order and reveal robustness to certain input-data misspecifications. Collectively, the results offer a principled account of why over-parameterized networks generalize well by preferentially fitting simple components early in training, and they quantify how sample size and network width influence the learnability of different frequencies.
Abstract
An intriguing phenomenon observed during training neural networks is the spectral bias, which states that neural networks are biased towards learning less complex functions. The priority of learning functions with low complexity might be at the core of explaining generalization ability of neural network, and certain efforts have been made to provide theoretical explanation for spectral bias. However, there is still no satisfying theoretical result justifying the underlying mechanism of spectral bias. In this paper, we give a comprehensive and rigorous explanation for spectral bias and relate it with the neural tangent kernel function proposed in recent work. We prove that the training process of neural networks can be decomposed along different directions defined by the eigenfunctions of the neural tangent kernel, where each direction has its own convergence rate and the rate is determined by the corresponding eigenvalue. We then provide a case study when the input data is uniformly distributed over the unit sphere, and show that lower degree spherical harmonics are easier to be learned by over-parameterized neural networks. Finally, we provide numerical experiments to demonstrate the correctness of our theory. Our experimental results also show that our theory can tolerate certain model misspecification in terms of the input data distribution.