Optimal Information Blending with Measurements in the L² Sphere

Abstract

We consider a sequential information collection problem where a risk-averse decision-maker updates a Bayesian belief about the unknown objective function of a linear program. The information is collected in the form of a linear combination of the objective coefficients, subject to random noise. We have the ability to choose the weights in the linear combination, creating a new, nonconvex continuous optimization problem, which we refer to as information blending. We develop two optimal blending strategies: an active learning method that maximizes uncertainty reduction, and an economic approach that maximizes an expected improvement criterion. Semidefinite programming relaxations are used to create efficient convex approximations to the nonconvex blending problem.

Optimal Learning in Linear Regression with Combinatorial Feature Selection

“…By contrast, in regression, we only observe a single scalar response for the chosen set of features, thus creating correlations. Another recent work by Defourny et al (2015) has considered semidefinite programming (SDP) relaxations for VIPs, but assumes known sampling noise as well as a continuous decision space. Finally, one stream of research (Norkin et al 1998, Xu andNelson 2013) applies branch-and-bound techniques to discrete simulation optimization, but treats the objective function as a black box, without the additional parametric structure afforded by regression.…”