How many Neurons do we need? A refined Analysis for Shallow Networks trained with Gradient Descent

We analyze the generalization properties of two-layer neural networks in the neural tangent kernel (NTK) regime, trained with gradient descent (GD). For early stopped GD we derive fast rates of convergence that are known to be minimax optimal in the framework of non-parametric regression in reproducing kernel Hilbert spaces. On our way, we precisely keep track of the number of hidden neurons required for generalization and improve over existing results. We further show that the weights during training remain in a vicinity around initialization, the radius being dependent on structural assumptions such as degree of smoothness of the regression function and eigenvalue decay of the integral operator associated to the NTK.

Random feature approximation for general spectral methods

Random feature approximation is arguably one of the most popular techniques to speed up kernel methods in large scale algorithms and provides a theoretical approach to the analysis of deep neural networks. We analyze generalization properties for a large class of spectral regularization methods combined with random features, containing kernel methods with implicit regularization such as gradient descent or explicit methods like Tikhonov regularization. For our estimators we obtain optimal learning rates over regularity classes (even for classes that are not included in the reproducing kernel Hilbert space), which are defined through appropriate source conditions. This improves or completes previous results obtained in related settings for specific kernel algorithms.