Gaining Outlier Resistance with Progressive Quantiles: Fast Algorithms and Theoretical Studies
This addresses the problem of outlier sensitivity in statistical estimation for big-data applications, offering a robust solution that is incremental in improving computational efficiency and theoretical guarantees.
The paper introduces an outlier-resistant estimation framework to robustify arbitrary loss functions, achieving minimax rate optimality in both low and high dimensions, with experiments showing excellent performance in regression, classification, and neural networks in the presence of gross outliers.
Outliers widely occur in big-data applications and may severely affect statistical estimation and inference. In this paper, a framework of outlier-resistant estimation is introduced to robustify an arbitrarily given loss function. It has a close connection to the method of trimming and includes explicit outlyingness parameters for all samples, which in turn facilitates computation, theory, and parameter tuning. To tackle the issues of nonconvexity and nonsmoothness, we develop scalable algorithms with implementation ease and guaranteed fast convergence. In particular, a new technique is proposed to alleviate the requirement on the starting point such that on regular datasets, the number of data resamplings can be substantially reduced. Based on combined statistical and computational treatments, we are able to perform nonasymptotic analysis beyond M-estimation. The obtained resistant estimators, though not necessarily globally or even locally optimal, enjoy minimax rate optimality in both low dimensions and high dimensions. Experiments in regression, classification, and neural networks show excellent performance of the proposed methodology at the occurrence of gross outliers.