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

• URL: http://arxiv.org/abs/2007.00884v2
• Date: Thu, 6 Aug 2020 23:30:27 GMT
• Title: A Revision of Neural Tangent Kernel-based Approaches for Neural Networks
• Authors: Kyung-Su Kim, Aur\'elie C. Lozano, Eunho Yang
• Abstract summary: We use the neural tangent kernel to show that networks can fit any finite training sample perfectly. A simple and analytic kernel function was derived as indeed equivalent to a fully-trained network. Our tighter analysis resolves the scaling problem and enables the validation of the original NTK-based results.
• Score: 34.75076385561115
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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