Abstract. Our goal is to build robust optimization problems for making decisions based on complex data from the past. In robust optimization (RO) generally, the goal is to create a policy for decision-making that is robust to our uncertainty about the future. In particular, we want our policy to best handle the the worst possible situation that could arise, out of an uncertainty set of possible situations. Classically, the uncertainty set is simply chosen by the user, or it might be estimated in overly simplistic ways with strong assumptions; whereas in this work, we learn the uncertainty set from data collected in the past. The past data are drawn randomly from an (unknown) possibly complicated high-dimensional distribution. We propose a new uncertainty set design and show how tools from statistical learning theory can be employed to provide probabilistic guarantees on the robustness of the policy.

Abstract. In this paper, we consider a supervised learning setting where side knowledge is provided about the labels of unlabeled examples. The side knowledge has the effect of reducing the hypothesis space, leading to tighter generalization bounds, and thus possibly better generalization. We consider several types of side knowledge, the first leading to linear and polygonal constraints on the hypothesis space, the second leading to quadratic constraints, and the last leading to conic constraints. We show how different types of domain knowledge can lead directly to these kinds of side knowledge. We prove bounds on complexity measures of the hypothesis space for quadratic and conic side knowledge, and show that these bounds are tight in a specific sense for the quadratic case. 1.