Frequency Bias in Neural Networks for Input of Non-Uniform Density
This addresses the problem of understanding training dynamics in realistic datasets for researchers in machine learning theory, but it is incremental as it builds on prior work on frequency bias and NTK models.
The paper investigates how non-uniform data density affects the frequency bias in neural networks, showing that convergence time for learning a harmonic function scales as O(κ^d/p(x)), where p(x) is local density, with analytical and empirical results for shallow and deep networks.
Recent works have partly attributed the generalization ability of over-parameterized neural networks to frequency bias -- networks trained with gradient descent on data drawn from a uniform distribution find a low frequency fit before high frequency ones. As realistic training sets are not drawn from a uniform distribution, we here use the Neural Tangent Kernel (NTK) model to explore the effect of variable density on training dynamics. Our results, which combine analytic and empirical observations, show that when learning a pure harmonic function of frequency $κ$, convergence at a point $\x \in \Sphere^{d-1}$ occurs in time $O(κ^d/p(\x))$ where $p(\x)$ denotes the local density at $\x$. Specifically, for data in $\Sphere^1$ we analytically derive the eigenfunctions of the kernel associated with the NTK for two-layer networks. We further prove convergence results for deep, fully connected networks with respect to the spectral decomposition of the NTK. Our empirical study highlights similarities and differences between deep and shallow networks in this model.