This paper primarily investigates the convergence of the Neural Tangent Kernel (NTK) in classification problems. This study firstly show the strictly positive definiteness of NTK of multi-layer fully connected neural networks and residual neural networks. Then, through a contradiction argument, it indicates that, during training with the cross-entropy loss function, the neural network parameters diverge due to the strictly positive definiteness of the NTK. Consequently, the empirical NTK does not consistently converge but instead diverges as time approaches infinity. This finding implies that NTK theory is not applicable in this context, highlighting significant theoretical implications for the study of neural networks in classification problems. These results can also be easily generalized to other network structures, provided that the NTK is strictly positive definite.

This paper aims to discuss the impact of random initialization of neural networks in the neural tangent kernel (NTK) theory, which is ignored by most recent works in the NTK theory. It is well known that as the network's width tends to infinity, the neural network with random initialization converges to a Gaussian process \(f^{\mathrm{GP}}\), which takes values in \(L^{2}(\mathcal{X})\), where \(\mathcal{X}\) is the domain of the data. In contrast, to adopt the traditional theory of kernel regression, most recent works introduced a special mirrored architecture and a mirrored (random) initialization to ensure the network's output is identically zero at initialization. Therefore, it remains a question whether the conventional setting and mirrored initialization would make wide neural networks exhibit different generalization capabilities. In this paper, we first show that the training dynamics of the gradient flow of neural networks with random initialization converge uniformly to that of the corresponding NTK regression with random initialization \(f^{\mathrm{GP}}\). We then show that \(\mathbf{P}(f^{\mathrm{GP}} \in [\mathcal{H}^{\mathrm{NT}}]^{s}) = 1\) for any \(s < \frac{3}{d+1}\) and \(\mathbf{P}(f^{\mathrm{GP}} \in [\mathcal{H}^{\mathrm{NT}}]^{s}) = 0\) for any \(s \geq \frac{3}{d+1}\), where \([\mathcal{H}^{\mathrm{NT}}]^{s}\) is the real interpolation space of the RKHS \(\mathcal{H}^{\mathrm{NT}}\) associated with the NTK. Consequently, the generalization error of the wide neural network trained by gradient descent is \(\Omega(n^{-\frac{3}{d+3}})\), and it still suffers from the curse of dimensionality. Thus, the NTK theory may not explain the superior performance of neural networks.

In this paper, we provide a strategy to determine the eigenvalue decay rate (EDR) of a large class of kernel functions defined on a general domain rather than $\mathbb{S}^{d}$. This class of kernel functions include but are not limited to the neural tangent kernel associated with neural networks with different depths and various activation functions. After proving that the dynamics of training the wide neural networks uniformly approximated that of the neural tangent kernel regression on general domains, we can further illustrate the minimax optimality of the wide neural network provided that the underground truth function $f\in [\mathcal H_{\mathrm{NTK}}]^{s}$, an interpolation space associated with the RKHS $\mathcal{H}_{\mathrm{NTK}}$ of NTK. We also showed that the overfitted neural network can not generalize well. We believe our approach for determining the EDR of kernels might be also of independent interests.