We introduce a Banach space-valued extension of random feature learning, a data-driven supervised machine learning technique for large-scale kernel approximation. By randomly initializing the feature maps, only the linear readout needs to be trained, which reduces the computational complexity substantially. Viewing random feature models as Banach space-valued random variables, we prove a universal approximation result in the corresponding Bochner space. Moreover, we derive approximation rates and an explicit algorithm to learn an element of the given Banach space by such models. The framework of this paper includes random trigonometric/Fourier regression and in particular random neural networks which are single-hidden-layer feedforward neural networks whose weights and biases are randomly initialized, whence only the linear readout needs to be trained. For the latter, we can then lift the universal approximation property of deterministic neural networks to random neural networks, even within function spaces over non-compact domains, e.g., weighted spaces, $L^p$-spaces, and (weighted) Sobolev spaces, where the latter includes the approximation of the (weak) derivatives. In addition, we analyze when the training costs for approximating a given function grow polynomially in both the input/output dimension and the reciprocal of a pre-specified tolerated approximation error. Furthermore, we demonstrate in a numerical example the empirical advantages of random feature models over their deterministic counterparts.

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