  We propose a new formulation of robust regression by integrating all
realizations of the uncertainty set and taking an averaged approach to obtain
the optimal solution for the ordinary least-squared regression problem. We show
that this formulation surprisingly recovers ridge regression and establishes
the missing link between robust optimization and the mean squared error
approaches for existing regression problems. We first prove the equivalence for
four uncertainty sets: ellipsoidal, box, diamond, and budget, and provide
closed-form formulations of the penalty term as a function of the sample size,
feature size, as well as perturbation protection strength. We then show in
synthetic datasets with different levels of perturbations, a consistent
improvement of the averaged formulation over the existing worst-case
formulation in out-of-sample performance. Importantly, as the perturbation
level increases, the improvement increases, confirming our method's advantage
in high-noise environments. We report similar improvements in the out-of-sample
datasets in real-world regression problems obtained from UCI datasets.
