[Paper Review] Towards Understanding the Spectral Bias of Deep Learning

Yuan Cao, Zhiying Fang+3 more|arXiv (Cornell University)|Dec 3, 2019

Stochastic Gradient Optimization Techniques57 references64 citations

TL;DR

The paper analyzes spectral bias through the neural tangent kernel (NTK), proving that gradient descent converges along NTK eigenfunctions at rates tied to their eigenvalues, with a case study on uniform sphere data showing low-degree spherical harmonics are learned faster, backed by experiments.

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.

Motivation & Objective

• Motivate and formalize spectral bias in over-parameterized neural networks.
• Link spectral bias to the neural tangent kernel and its eigenstructure.
• Provide general convergence results along NTK eigendirections.
• Characterize NTK spectra for uniform sphere data and relate to learning of low-degree harmonics.
• Validate theoretical findings with numerical experiments across settings.

Proposed method

• Model neural networks in the neural tangent kernel (NTK) regime and derive gradient-descent dynamics along NTK eigenfunctions.
• Define the NTK for two-layer ReLU networks and express it as a sum of arc-cosine kernels of degree 0 and 1.
• Introduce the integral operator L_kappa and its eigenfunctions/eigenvalues to describe convergence directions.
• Prove a generic theorem that convergence along the NTK eigendirections depends on corresponding eigenvalues under certain sample and width conditions.
• Perform a spectral analysis of NTK under uniform spherical data, obtaining explicit eigenvalues/eigenfunctions (spherical harmonics) and their decay rates.
• Provide corollaries giving explicit convergence rates for uniform sphere data, tying learning speed to eigenvalues.

Experimental results

Research questions

• RQ1 How does gradient descent on over-parameterized networks behave along NTK eigen-directions?
• RQ2 How do NTK eigenvalues control the convergence rate in learning different frequency components of the target function?
• RQ3 What is the spectrum of the NTK when inputs are uniformly distributed on the unit sphere, and how does it relate to spherical harmonics?
• RQ4 Can low-degree components be learned faster than high-degree ones under realistic width and sample-size conditions?

Key findings

• Training error projections onto NTK eigenspaces converge at rates determined by the corresponding eigenvalues.
• Lower-frequency components (larger NTK eigenvalues) are learned faster and with narrower networks fewer samples.
• When data are uniformly distributed on the sphere, NTK eigenfunctions align with spherical harmonics and eigenvalues decay as mu_k = Omega(k^{-d-1}) for k >> d or Omega(d^{-k+1}) for d >> k.
• The theory holds for arbitrary target functions, not requiring the target to lie in the NTK-induced RKHS.
• Experiments on learning combinations of spherical harmonics and simple functions corroborate the projected residual convergence rates.
• The results tolerate certain model misspecifications in data distribution.