  To optimize efficiently over discrete data and with only few available target
observations is a challenge in Bayesian optimization. We propose a continuous
relaxation of the objective function and show that inference and optimization
can be computationally tractable. We consider in particular the optimization
domain where very few observations and strict budgets exist; motivated by
optimizing protein sequences for expensive to evaluate bio-chemical properties.
The advantages of our approach are two-fold: the problem is treated in the
continuous setting, and available prior knowledge over sequences can be
incorporated directly. More specifically, we utilize available and learned
distributions over the problem domain for a weighting of the Hellinger distance
which yields a covariance function. We show that the resulting acquisition
function can be optimized with both continuous or discrete optimization
algorithms and empirically assess our method on two bio-chemical sequence
optimization tasks.
