Methodological advances in structured statistical learning

Abstract

This dissertation explores methodological advances in statistical learning, specifically addressing two fundamental challenges: the interpretability of complex association structures in multivariate categorical data analysis, and privacy-preserving distributed inference across heterogeneous datasets. First, we propose a penalized likelihood framework tailored for multivariate categorical response regression, encompassing classical discrete graphical models such as the Ising model, Potts model, and hypergraph models. Utilizing a distinctive subspace decomposition, the method explicitly captures mutual, joint, and conditionally independent associations between categorical variables, facilitating interpretable representations of association structures. We derive theoretical guarantees, establishing error bounds that hold particularly in high-dimensional contexts. Comprehensive simulation studies demonstrate that our approach achieves greater interpretability and improved predictive accuracy compared to existing methods. Second, we tackle data-sharing challenges under stringent privacy constraints and site heterogeneity, common in multi-site clinical trials. We introduce a robust distributed algorithm for high-dimensional inference and structure learning. Our heterogeneous model integrates global and site-specific effects, employing nonconvex regularization via difference of convex programming under an l0 constraint, ensuring selection consistency and computational feasibility. Despite the underlying optimization being NP-hard, our method converges globally in polynomial time under realistic conditions. By exclusively penalizing nuisance parameters, our approach maintains valid statistical inference, directly addressing practical data-sharing constraints.