Title: Dealing with Integer-valued Variables in Bayesian Optimization with Gaussian Processes,
Abstract: Bayesian optimization (BO) methods are useful for optimizing functions that
are expensive to evaluate, lack an analytical expression and whose evaluations
can be contaminated by noise. These methods rely on a probabilistic model of
the objective function, typically a Gaussian process (GP), upon which an
acquisition function is built. This function guides the optimization process
and measures the expected utility of performing an evaluation of the objective
at a new point. GPs assume continous input variables. When this is not the
case, such as when some of the input variables take integer values, one has to
introduce extra approximations. A common approach is to round the suggested
variable value to the closest integer before doing the evaluation of the
objective. We show that this can lead to problems in the optimization process
and describe a more principled approach to account for input variables that are
integer-valued. We illustrate in both synthetic and a real experiments the
utility of our approach, which significantly improves the results of standard
BO methods on problems involving integer-valued variables.