  In this paper, we aim at developing scalable neural network-type learning
systems. Motivated by the idea of "constructive neural networks" in
approximation theory, we focus on "constructing" rather than "training"
feed-forward neural networks (FNNs) for learning, and propose a novel FNNs
learning system called the constructive feed-forward neural network (CFN).
Theoretically, we prove that the proposed method not only overcomes the
classical saturation problem for FNN approximation, but also reaches the
optimal learning rate when the regression function is smooth, while the
state-of-the-art learning rates established for traditional FNNs are only near
optimal (up to a logarithmic factor). A series of numerical simulations are
provided to show the efficiency and feasibility of CFN via comparing with the
well-known regularized least squares (RLS) with Gaussian kernel and extreme
learning machine (ELM).
