A Theory of Neural Tangent Kernel Alignment and Its Influence on Training

The training dynamics and generalization properties of neural networks (NN) can be precisely characterized in function space via the neural tangent kernel (NTK). Structural changes to the NTK during training reflect feature learning and underlie the superior performance of networks outside of the static kernel regime. In this work, we seek to theoretically understand kernel alignment, a prominent and ubiquitous structural change that aligns the NTK with the target function. We first study a toy model of kernel evolution in which the NTK evolves to accelerate training and show that alignment naturally emerges from this demand. We then study alignment mechanism in deep linear networks and two layer ReLU networks. These theories provide good qualitative descriptions of kernel alignment and specialization in practical networks and identify factors in network architecture and data structure that drive kernel alignment. In nonlinear networks with multiple outputs, we identify the phenomenon of kernel specialization, where the kernel function for each output head preferentially aligns to its own target function. Together, our results provide a mechanistic explanation of how kernel alignment emerges during NN training and a normative explanation of how it benefits training.