  We propose a sequential learning policy for noisy discrete global
optimization and ranking and selection (R\&S) problems with high dimensional
sparse belief functions, where there are hundreds or even thousands of
features, but only a small portion of these features contain explanatory power.
We aim to identify the sparsity pattern and select the best alternative before
the finite budget is exhausted. We derive a knowledge gradient policy for
sparse linear models (KGSpLin) with group Lasso penalty. This policy is a
unique and novel hybrid of Bayesian R\&S with frequentist learning.
Particularly, our method naturally combines B-spline basis expansion and
generalizes to the nonparametric additive model (KGSpAM) and functional ANOVA
model. Theoretically, we provide the estimation error bounds of the posterior
mean estimate and the functional estimate. Controlled experiments show that the
algorithm efficiently learns the correct set of nonzero parameters even when
the model is imbedded with hundreds of dummy parameters. Also it outperforms
the knowledge gradient for a linear model.
