A Unified Robust Regression Model for Lasso-like Algorithms

We develop a unified robust linear regression model and show that it is equivalent to a general regularization framework to encourage sparse-like structure that contains group Lasso and fused Lasso as specific examples. This provides a robustness interpretation of these widely applied Lasso-like algorithms, and allows us to construct novel generalizations of Lasso-like algorithms by considering different uncertainty sets. Using this robustness interpretation, we present new sparsity results, and establish the statistical consistency of the proposed regularized linear regression. This work extends a classical result from Xu et al. (2010) that relates standard Lasso with robust linear regression to learning problems with more general sparse-like structures, and provides new robustness-based tools to to understand learning problems with sparse-like structures.