Topics in Nonparametric Machine Learning: Subgroup Analysis and Deep Neural Networks Regression
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Topics in Nonparametric Machine Learning: Subgroup Analysis and Deep Neural Networks Regression


In recent years, modern technology has facilitated the collection of large-scale data from medical records, health insurance databases, and other platforms. Due to the complex structure, the analysis of such data is very challenging. The dissertation focuses on the nonparametric machine learning techniques in subgroup analysis and deep neural networks regression.

The first part of the dissertation studies the heterogeneity in the disease progression, which is essential to the development of precision medicine that aims to tailor treatments to subgroups of patients with similar characteristics. Without a priori knowledge of grouping information, our goal is to identify subgroups of individuals who share a common disorder progress over time, i.e. longitudinal trajectory. We develop a subject-specific nonparametric regression model, where the heterogeneous trajectories are modeled through the subject-specific unknown functions and can be approximated by B-splines. We then apply the fusion penalized method that can automatically divide the individuals into different subgroups based on the B-spline coefficients as well as estimating the coefficients simultaneously. We also illustrate the performance of this method through simulation studies and a biomedical data application.

The second part of the dissertation considers a sparse deep ReLU network (SDRN) estimator obtained from empirical risk minimization with a Lipschitz loss function in the presence of a large number of features. Instead of utilizing full grids, the unknown target function is approximated by a deep ReLU network with sparse grids. Our framework can be applied to a variety of regression and classification problems. The unknown target function to estimate is assumed to be in a Sobolev space with mixed derivatives. Functions in this space only need to satisfy a smoothness condition rather than having a compositional structure. We develop non-asymptotic excess riskbounds for our SDRN estimator. We further derive that the SDRN estimator can achieve the same minimax rate of estimation (up to logarithmic factors) as one-dimensional nonparametric regression when the dimension of the features is fixed, and the estimator has a suboptimal rate when the dimension grows with the sample size. We show that the depth and the total number of nodes and weights of the ReLU network need to grow as the sample size increases to ensure a good performance, and also investigate how fast they should increase with the sample size. These results provide an important theoretical guidance and basis for empirical studies by deep neural networks.

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