Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operators

In this work, we investigate the generalization properties of random feature methods. Our analysis extends prior results for Tikhonov regularization to a broad class of spectral regularization techniques and further generalizes the setting to operator-valued kernels. This unified framework enables a rigorous theoretical analysis of neural operators and neural networks through the lens of the Neural Tangent Kernel (NTK). In particular, it allows us to establish optimal learning rates and provides a good understanding of how many neurons are required to achieve a given accuracy. Furthermore, we establish minimax rates in the well-specified case and also in the misspecified case, where the target is not contained in the reproducing kernel Hilbert space. These results sharpen and complete earlier findings for specific kernel algorithms.

Optimal Convergence Rates for Neural Operators

We introduce the neural tangent kernel (NTK) regime for two-layer neural operators and analyze their generalization properties. For early-stopped gradient descent (GD), we derive fast convergence rates that are known to be minimax optimal within the framework of non-parametric regression in reproducing kernel Hilbert spaces (RKHS). We provide bounds on the number of hidden neurons and the number of second-stage samples necessary for generalization. To justify our NTK regime, we additionally show that any operator approximable by a neural operator can also be approximated by an operator from the RKHS. A key application of neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider the standard Poisson equation to illustrate our theoretical findings with simulations.

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.