Optimal Learning in Linear Regression with Combinatorial Feature Selection

We present a new framework for sequential information collection in applications where regression is used to learn about a set of unknown parameters and alternates with optimization to design new data points. Such problems can be handled using the framework of ranking and selection (R&S), but traditional R&S procedures will experience high computational costs when the decision space grows combinatorially. This challenge arises in many applications of business analytics; in particular, we are motivated by the problem of efficiently learning effective strategies for nonprofit fundraising. We present a value of information procedure for simultaneously learning unknown regression parameters and unknown sampling noise. We then develop approximate versions of the procedure, based on optimal quantization, that retain good performance and scale better to large problems.