A Revision of Neural Tangent Kernel-based Approaches for Neural Networks

Recent theoretical works based on the neural tangent kernel (NTK) have shed light on the optimization and generalization of over-parameterized networks, and partially bridge the gap between their practical success and classical learning theory. Especially, using the NTK-based approach, the following three representative results were obtained: (1) A training error bound was derived to show that networks can fit any finite training sample perfectly by reflecting a tighter characterization of training speed depending on the data complexity. (2) A generalization error bound invariant of network size was derived by using a data-dependent complexity measure (CMD). It follows from this CMD bound that networks can generalize arbitrary smooth functions. (3) A simple and analytic kernel function was derived as indeed equivalent to a fully-trained network. This kernel outperforms its corresponding network and the existing gold standard, Random Forests, in few shot learning. For all of these results to hold, the network scaling factor $\kappa$ should decrease w.r.t. sample size n. In this case of decreasing $\kappa$, however, we prove that the aforementioned results are surprisingly erroneous. It is because the output value of trained network decreases to zero when $\kappa$ decreases w.r.t. n. To solve this problem, we tighten key bounds by essentially removing $\kappa$-affected values. Our tighter analysis resolves the scaling problem and enables the validation of the original NTK-based results.