# Gaining Outlier Resistance with Progressive Quantiles: Fast Algorithms and Theoretical Studies

arXiv: https://arxiv.org/abs/2112.08471 · HTML: https://mlsanity.com/paper/2112.08471/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 60 references that are in the map.
- First posted on arXiv 15 Dec 2021.

## Builds on (most cited first, 6 of 6)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Statistical consistency and asymptotic normality for high-dimensional robust M-estimators](https://mlsanity.com/paper/1501.00312/) (2015)
- [Robust Reduced Rank Regression](https://mlsanity.com/paper/1509.03938/) (2015)
- [On Cross-validation for Sparse Reduced Rank Regression](https://mlsanity.com/paper/1812.11555/) (2018)
- [Robust and sparse estimation methods for high dimensional linear and logistic regression](https://mlsanity.com/paper/1703.04951/) (2017)
- [Penalized Maximum Tangent Likelihood Estimation and Robust Variable Selection](https://mlsanity.com/paper/1708.05439/) (2017)

## Built on by (most cited first, 5 of 5)

- [Analysis of Generalized Bregman Surrogate Algorithms for Nonsmooth Nonconvex Statistical Learning](https://mlsanity.com/paper/2112.09191/) (2021)
- [The Cellwise Minimum Covariance Determinant Estimator](https://mlsanity.com/paper/2207.13493/) (2022)
- [Slow Kill for Big Data Learning](https://mlsanity.com/paper/2305.01726/) (2023)
- [Mind the Jumps: A Scalable Robust Local Gaussian Process for Multidimensional Response Surfaces with Discontinuities](https://mlsanity.com/paper/2512.12574/) (2025)
- [Robust penalized least squares of depth trimmed residuals regression for high-dimensional data](https://mlsanity.com/paper/2309.01666/) (2023)

## Abstract (arXiv)

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.