General Frequency Principle for Linear Approximation Classes

Yujun Wang, Yiqi Gu

In this paper, we study a general frequency principle under gradient descent training for the finite linear approximation classes. Under suitable assumptions, we prove that the learner amplitude relative to the target amplitude vanishes at high frequencies, uniformly throughout training. We also establish sufficient conditions for low-frequency priority, covering single-frequency targets trained separately and two frequency components trained in the same run. This general frequency principle does not imply that lower frequencies always converge faster. Instead, for two frequencies that satisfy certain comparison conditions, one component will take longer to learn than the other; and in common cases, the longer component has higher frequency. Numerical experiments with Legendre and trigonometric bases validate the theoretical results.