Distributed Learning, Prediction and Detection in Probabilistic Graphs.

Abstract

Critical to high-dimensional statistical estimation is to exploit the structure in the data distribution. Probabilistic graphical models provide an efficient framework for representing complex joint distributions of random variables through their conditional dependency graph, and can be adapted to many high-dimensional machine learning applications.

This dissertation develops the probabilistic graphical modeling technique for three statistical estimation problems arising in real-world applications: distributed and parallel learning in networks, missing-value prediction in recommender systems, and emerging topic detection in text corpora. The common theme behind all proposed methods is a combination of parsimonious representation of uncertainties in the data, optimization surrogate that leads to computationally efficient algorithms, and fundamental limits of estimation performance in high dimension.

More specifically, the dissertation makes the following theoretical contributions:

(1) We propose a distributed and parallel framework for learning the parameters in Gaussian graphical models that is free of iterative global message passing. The proposed distributed estimator is shown to be asymptotically consistent, improve with increasing local neighborhood sizes, and have a high-dimensional error rate comparable to that of the centralized maximum likelihood estimator.

(2) We present a family of latent variable Gaussian graphical models whose marginal precision matrix has a “low-rank plus sparse” structure. Under mild conditions, we analyze the high-dimensional parameter error bounds for learning this family of models using regularized maximum likelihood estimation.

(3) We consider a hypothesis testing framework for detecting emerging topics in topic models, and propose a novel surrogate test statistic for the standard likelihood ratio. By leveraging the theory of empirical processes, we prove asymptotic consistency for the proposed test and provide guarantees of the detection performance.