On the Exact Computation of Linear Frequency Principle Dynamics and Its Generalization

Recent works show the intriguing phenomenon of the frequency principle (F-Principle) that deep neural networks (DNNs) fit the target function from low to high frequency during training, which provides insight into the training and generalization behavior of DNNs in complex tasks. In this paper, through analysis of an infinite-width two-layer NN in the neural tangent kernel regime, we derive the exact differential equation, namely the linear frequency-principle (LFP) model, governing the evolution of NN output function in the frequency domain during training. Our exact computation applies for general activation functions with no assumption on size and distribution of training data. This LFP model unravels that higher frequencies evolve polynomially or exponentially slower than lower frequencies depending on the smoothness/regularity of the activation function. We further bridge the gap between training dynamics and generalization by proving that the LFP model implicitly minimizes a frequency-principle norm (FP-norm) of the learned function, by which higher frequencies are more severely penalized depending on the inverse of their evolution rate. Finally, we derive an a priori generalization error bound controlled by the FP-norm of the target function, which provides a theoretical justification for the empirical results that DNNs often generalize well for low-frequency functions.