Two-Layer Neural Networks with Values in a Banach Space

We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped a partial order, i.e. it Riesz space. As nonlinearity choose lattice operation of taking positive part; in case $\mathbb R^d$-valued this corresponds to ReLU activation function. prove inverse direct approximation theorems Monte-Carlo rates, extending existing results for finite-dimensional case. second part paper, consider training such using finite amount noisy observations from regularisation theory viewpoint. discuss regularity conditions known as source obtain convergence rates Bregman distance regime when both noise level goes zero number samples infinity at appropriate rates.