  Suppose there is a large collection of items, each with an associated cost
and an inherent utility that is revealed only once we commit to selecting it.
Given a budget on the cumulative cost of the selected items, how can we pick a
subset of maximal value? This task generalizes several important problems such
as multi-arm bandits, active search and the knapsack problem. We present an
algorithm, GP-Select, which utilizes prior knowledge about similarity be- tween
items, expressed as a kernel function. GP-Select uses Gaussian process
prediction to balance exploration (estimating the unknown value of items) and
exploitation (selecting items of high value). We extend GP-Select to be able to
discover sets that simultaneously have high utility and are diverse. Our
preference for diversity can be specified as an arbitrary monotone submodular
function that quantifies the diminishing returns obtained when selecting
similar items. Furthermore, we exploit the structure of the model updates to
achieve an order of magnitude (up to 40X) speedup in our experiments without
resorting to approximations. We provide strong guarantees on the performance of
GP-Select and apply it to three real-world case studies of industrial
relevance: (1) Refreshing a repository of prices in a Global Distribution
System for the travel industry, (2) Identifying diverse, binding-affine
peptides in a vaccine de- sign task and (3) Maximizing clicks in a web-scale
recommender system by recommending items to users.
