Gaining Outlier Resistance With Progressive Quantiles: Fast Algorithms and Theoretical Studies
提出一种抗离群值的估计框架，通过引入显式离群参数来鲁棒化任意损失函数，开发了可扩展的快速收敛算法，并在低维和高维下达到极小极大最优性。
Outliers widely occur in big-data applications and may severely affect statistical estimation and inference. In this article, 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. Supplementary materials for this article are available online.