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On regularized square-root Regression Problems: Distributionally Robust Interpretation and Fast Computations

JOURNAL OF MACHINE LEARNING RESEARCH 2022年第1期23卷 1-39页

作者： Chu, Hong T. M. Toh, Kim-Chuan Zhang, Yangjing Natl Univ Singapore Dept Math Singapore 119076 Singapore Natl Univ Singapore Inst Operat Res & Analyt Singapore 119076 Singapore Chinese Acad Sci Acad Math & Syst Sci Inst Appl Math Beijing 100190 Peoples R China

square-root (loss) regularized models have recently become popular in linear regression due to their nice statistical properties. Moreover, some of these models can be interpreted as the distributionally robust optimization counterparts of the traditional least-squares regularized models. In this paper, we give a unified proof to show that any square-root regularized model whose penalty function being the sum of a simple norm and a seminorm can be interpreted as the distributionally robust optimization (DRO) formulation of the corresponding least-squares problem. In particular, the optimal transport cost in the DRO formulation is given by a certain dual form of the penalty. To solve the resulting square -root regularized model whose loss function and penalty function are both nonsmooth, we design a proximal point dual semismooth Newton algorithm and demonstrate its efficiency when the penalty is the sparse group Lasso penalty or the fused Lasso penalty. Extensive experiments demonstrate that our algorithm is highly efficient for solving the square-root sparse group Lasso problems and the square-root fused Lasso problems.

关键词： square-root regularized model distributionally robust optimization proximal point algorithm semismooth Newton group Lasso

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On regularized square-root regression problems: distributionally robust interpretation and fast computations

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2022年第1期23卷 13885-13923页

作者： Hong T.M. Chu Kim-Chuan Toh Yangjing Zhang Department of Mathematics National University of Singapore Singapore Department of Mathematics and Institute of Operations Research and Analytics National University of Singapore Singapore Institute of Applied Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences People's Republic of China

square-root (loss) regularized models have recently become popular in linear regression due to their nice statistical properties. Moreover, some of these models can be interpreted as the distributionally robust optimization counterparts of the traditional least-squares regularized models. In this paper, we give a unified proof to show that any square-root regularized model whose penalty function being the sum of a simple norm and a seminorm can be interpreted as the distributionally robust optimization (DRO) formulation of the corresponding least-squares problem. In particular, the optimal transport cost in the DRO formulation is given by a certain dual form of the penalty. To solve the resulting square-root regularized model whose loss function and penalty function are both nonsmooth, we design a proximal point dual semismooth Newton algorithm and demonstrate its efficiency when the penalty is the sparse group Lasso penalty or the fused Lasso penalty. Extensive experiments demonstrate that our algorithm is highly efficient for solving the square-root sparse group Lasso problems and the square-root fused Lasso problems.

关键词： square-root regularized model distributionally robust optimization proximal point algorithm semismooth Newton group Lasso

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