On Regularized Square-root Regression Problems: Distributionally Robust Interpretation and Fast Computations

TL;DR

This paper establishes a unified interpretation of square-root regularized models as distributionally robust optimization problems and introduces an efficient algorithm for solving these models with complex penalties.

Contribution

It provides a unified DRO interpretation for a class of square-root regularized models and develops a fast proximal point dual semismooth Newton algorithm for their efficient solution.

Findings

The proposed algorithm is highly efficient for sparse group Lasso problems.

The method effectively solves fused Lasso regularized models.

Experimental results confirm the computational advantages of the approach.

Abstract

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.