CIF: Small: Towards Robust Statistical Learning: Theory and Algorithms

University Of Southern California, Los Angeles CA

Investigators

Abstract

Machine learning algorithms are used to automate various tasks by finding patterns in the existing data. The mathematical analysis of machine learning algorithms starts by assuming that the available dataset is described by a model with certain properties. However, as real-world data often do not satisfy the model assumptions exactly, there is a need to reduce the gap between the "mathematical" and "real" worlds by weakening the mathematical assumptions. The concept of robustness plays a central role in understanding this gap. First, the project will formulate principles for building robust algorithms. The project will then apply these principles to address problems related to the existence of mathematically justified and computationally efficient robust methods for prediction and classification tasks, which are among the most popular problems solved by machine learning algorithms. The project will also support undergraduate research by training students to apply advanced methods to the analysis of modern data sets. Additional efforts will be made to establish closer ties between the academic and industry machine learning research communities. One part of the project is devoted to robust empirical risk minimization. Empirical risk minimization is one of the fundamental concepts underlying modern mathematical statistics and statistical learning algorithms, including regression and maximum likelihood estimation. However, empirical risk minimization is not robust in many scenarios, with a single "atypical point" amongst the observations possibly significantly affecting performance. The work done in the course of this project will lead to algorithms that avoid explicit outlier detection and removal, and which instead take advantage of existing or purposefully induced symmetries in the distribution of the data. The analysis of these new algorithms will require the development of novel techniques related to Bahadur-type representations of robust estimators, and of new generalizations of the median-of-means principle. Another part of the project aims at developing robust modifications of the Federated Learning algorithm, originally designed as a communication-effective alternative to the standard centralized datacenter framework. The project will design new and robust versions of the Federated Learning algorithm that provably work in the challenging scenario where the input data have different distributions. Finally, the investigator will address inferential problems in robust learning by devising robust versions of posterior distributions that are central objects in Bayesian statistics; he will study the Bernstein-von Mises theorem for these robust posteriors, a fundamental result connecting the frequentist and Bayesian methods. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.