A Theory of Non-Linear Feature Learning with One Gradient Step in Two-Layer Neural Networks
This work provides a theoretical foundation for understanding how gradient descent can learn non-linear features in neural networks, addressing a key bottleneck in deep learning theory.
The paper tackles the problem of non-linear feature learning in two-layer neural networks by showing that a growing learning rate enables the emergence of multiple polynomial features, leading to improved training and test errors as proven through rigorous analysis.
Feature learning is thought to be one of the fundamental reasons for the success of deep neural networks. It is rigorously known that in two-layer fully-connected neural networks under certain conditions, one step of gradient descent on the first layer can lead to feature learning; characterized by the appearance of a separated rank-one component -- spike -- in the spectrum of the feature matrix. However, with a constant gradient descent step size, this spike only carries information from the linear component of the target function and therefore learning non-linear components is impossible. We show that with a learning rate that grows with the sample size, such training in fact introduces multiple rank-one components, each corresponding to a specific polynomial feature. We further prove that the limiting large-dimensional and large sample training and test errors of the updated neural networks are fully characterized by these spikes. By precisely analyzing the improvement in the training and test errors, we demonstrate that these non-linear features can enhance learning.