Robust Regression and Lasso

We consider robust least-squares regression with feature-wise disturbance. We show that this formulation leads to tractable convex optimization problems, and we exhibit a particular uncertainty set for which the robust problem is equivalent to 1 regularized regression (Lasso). This provides an interpretation of Lasso from a robust optimization perspective. We generalize this robust formulation to consider more general uncertainty sets, which all lead to tractable convex optimization problems. Therefore, we provide a new methodology for designing regression algorithms, which generalize known formulations. The advantage is that robustness to disturbance is a physical property that can be exploited: in addition to obtaining new formulations, we use it directly to show sparsity properties of Lasso, as well as to prove a general consistency result for robust regression problems, including Lasso, from a unified robustness perspective.

• Type: Conference
• Year: 2008
• Where: NIPS
• Authors: Huan Xu, Constantine Caramanis, Shie Mannor
• Keywords: Convex Optimization Problems, Information Technology, NIPS 2008, Robust, Tractable Convex Optimization