A kernel-expanded stochastic neural network

The deep neural network suffers from many fundamental issues in machine learning. For example, it often gets trapped into a local minimum in training, and its prediction uncertainty is hard to be assessed. To address these issues, we propose the so-called kernel-expanded stochastic neural network (K-StoNet) model, which incorporates support vector regression as the first hidden layer and reformulates the neural network as a latent variable model. The former maps the input vector into an infinite dimensional feature space via a radial basis function kernel, ensuring the absence of local minima on its training loss surface. The latter breaks the high-dimensional non-convex neural network training problem into a series of low-dimensional convex optimization problems, and enables its prediction uncertainty easily assessed. The K-StoNet can be easily trained using the imputation-regularized optimization algorithm. Compared to traditional deep neural networks, K-StoNet possesses a theoretical guarantee to asymptotically converge to the global optimum and enables the prediction uncertainty easily assessed. The performances of the new model in training, prediction and uncertainty quantification are illustrated by simulated and real data examples.