The Annals of Statistics

Least quantile regression via modern optimization

Dimitris Bertsimas and Rahul Mazumder

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We address the Least Quantile of Squares (LQS) (and in particular the Least Median of Squares) regression problem using modern optimization methods. We propose a Mixed Integer Optimization (MIO) formulation of the LQS problem which allows us to find a provably global optimal solution for the LQS problem. Our MIO framework has the appealing characteristic that if we terminate the algorithm early, we obtain a solution with a guarantee on its sub-optimality. We also propose continuous optimization methods based on first-order subdifferential methods, sequential linear optimization and hybrid combinations of them to obtain near optimal solutions to the LQS problem. The MIO algorithm is found to benefit significantly from high quality solutions delivered by our continuous optimization based methods. We further show that the MIO approach leads to (a) an optimal solution for any dataset, where the data-points $(y_{i},\mathbf{x}_{i})$’s are not necessarily in general position, (b) a simple proof of the breakdown point of the LQS objective value that holds for any dataset and (c) an extension to situations where there are polyhedral constraints on the regression coefficient vector. We report computational results with both synthetic and real-world datasets showing that the MIO algorithm with warm starts from the continuous optimization methods solve small ($n=100$) and medium ($n=500$) size problems to provable optimality in under two hours, and outperform all publicly available methods for large-scale ($n=10,000$) LQS problems.

Article information

Ann. Statist. Volume 42, Number 6 (2014), 2494-2525.

First available in Project Euclid: 12 November 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J05: Linear regression 62G35: Robustness
Secondary: 90C11: Mixed integer programming 90C26: Nonconvex programming, global optimization

Least median of squares robust statistics least quantile regression algorithms mixed integer programming global optimization continuous optimization


Bertsimas, Dimitris; Mazumder, Rahul. Least quantile regression via modern optimization. Ann. Statist. 42 (2014), no. 6, 2494--2525. doi:10.1214/14-AOS1223.

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