Convex and Nonconvex Risk-Based Linear Regression at Scale

Can Wu (), Ying Cui (), Donghui Li () and Defeng Sun ()
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Can Wu: School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China; Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
Ying Cui: Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, Minnesota 55455
Donghui Li: School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
Defeng Sun: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

INFORMS Journal on Computing, 2023, vol. 35, issue 4, 797-816

Abstract: The value at risk (VaR) and the conditional value at risk (CVaR) are two popular risk measures to hedge against the uncertainty of data. In this paper, we provide a computational toolbox for solving high-dimensional sparse linear regression problems under either VaR or CVaR measures, the former being nonconvex and the latter convex. Unlike the empirical risk (neutral) minimization models in which the overall losses are decomposable across data, the aforementioned risk-sensitive models have nonseparable objective functions so that typical first order algorithms are not easy to scale. We address this scaling issue by adopting a semismooth Newton-based proximal augmented Lagrangian method of the convex CVaR linear regression problem. The matrix structures of the Newton systems are carefully explored to reduce the computational cost per iteration. The method is further embedded in a majorization–minimization algorithm as a subroutine to tackle the nonconvex VaR-based regression problem. We also discuss an adaptive sieving strategy to iteratively guess and adjust the effective problem dimension, which is particularly useful when a solution path associated with a sequence of tuning parameters is needed. Extensive numerical experiments on both synthetic and real data demonstrate the effectiveness of our proposed methods. In particular, they are about 53 times faster than the commercial package Gurobi for the CVaR-based sparse linear regression with 4,265,669 features and 16,087 observations.

Keywords: risk measures; (conditional) value-at-risk; sparsity; semismooth Newton; augmented Lagrangian; nonconvexity (search for similar items in EconPapers)
Date: 2023
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