Divergence of Empirical Neural Tangent Kernel in Classification Problems

This paper demonstrates that in classification problems, fully connected neural networks (FCNs) and residual neural networks (ResNets) cannot be approximated by kernel logistic regression based on the Neural Tangent Kernel (NTK) under overtraining (i.e., when training time approaches infinity). Specifically, when using the cross-entropy loss, regardless of how large the network width is (as long as it is finite), the empirical NTK diverges from the NTK on the training samples as training time increases. To establish this result, we first demonstrate the strictly positive definiteness of the NTKs for multi-layer FCNs and ResNets. Then, we prove that during training, % with the cross-entropy loss, the neural network parameters diverge if the smallest eigenvalue of the empirical NTK matrix (Gram matrix) with respect to training samples is bounded below by a positive constant. This behavior contrasts sharply with the lazy training regime commonly observed in regression problems. Consequently, using a proof by contradiction, we show that the empirical NTK does not uniformly converge to the NTK across all times on the training samples as the network width increases. We validate our theoretical results through experiments on both synthetic data and the MNIST classification task. This finding implies that NTK theory is not applicable in this context, with significant theoretical implications for understanding neural networks in classification problems.