Approximation analysis of outcome weighted learning by convolutional neural networks

Deep learning algorithms based on convolutional neural networks have witnessed a lot of practical successes in a wide variety of applications including computer vision and natural language processing. Meanwhile, outcome-weighted learning algorithms provide a general and popular framework to solve the problems of individualized treatment rule estimation and personalized dose finding arising from precision medicine, but most of the existing studies focus on linear models and kernel methods. With the availability of high-dimensional, massive, and complex data from patients' electronic health records, there is a rapidly increasing interest in integrating these two research topics. In this project, we plan to carry out rigorous mathematical analysis for outcome-weighted learning algorithms generated by deep convolutional neural networks. First, we propose a novel functional deep convolutional neural network with a rectified linear unit for the approximation of nonlinear and smooth functionals. Explicit rates of approximation will be derived to deepen our understanding of the approximation power of the proposed functional neural network and exploit its advantages compared to conventional methods. The cases when the target functional admits a special structure or the domain consists of analytical functions will also be considered. Then we shall establish the learning theory of functional deep convolutional neural networks by studying the consistency and error bounds of their generated deep functional learning algorithms. Finally, we shall derive error bounds and convergence rates for the outcome-weighted learning algorithms generated by deep convolutional neural networks when the covariates are high-dimensional vectors, random functions, or probability distributions, respectively.