  We address the problem of finding the maximizer of a nonlinear smooth
function, that can only be evaluated point-wise, subject to constraints on the
number of permitted function evaluations. This problem is also known as
fixed-budget best arm identification in the multi-armed bandit literature. We
introduce a Bayesian approach for this problem and show that it empirically
outperforms both the existing frequentist counterpart and other Bayesian
optimization methods. The Bayesian approach places emphasis on detailed
modelling, including the modelling of correlations among the arms. As a result,
it can perform well in situations where the number of arms is much larger than
the number of allowed function evaluation, whereas the frequentist counterpart
is inapplicable. This feature enables us to develop and deploy practical
applications, such as automatic machine learning toolboxes. The paper presents
comprehensive comparisons of the proposed approach, Thompson sampling,
classical Bayesian optimization techniques, more recent Bayesian bandit
approaches, and state-of-the-art best arm identification methods. This is the
first comparison of many of these methods in the literature and allows us to
examine the relative merits of their different features.
