  Bayesian optimization is a sample-efficient method for finding a global
optimum of an expensive-to-evaluate black-box function. A global solution is
found by accumulating a pair of query point and its function value, repeating
these two procedures: (i) modeling a surrogate function; (ii) maximizing an
acquisition function to determine where next to query. Convergence guarantees
are only valid when the global optimizer of the acquisition function is found
at each round and selected as the next query point. In practice, however, local
optimizers of an acquisition function are also used, since searching for the
global optimizer is often a non-trivial or time-consuming task. In this paper
we consider three popular acquisition functions, PI, EI, and GP-UCB induced by
Gaussian process regression. Then we present a performance analysis on the
behavior of local optimizers of those acquisition functions, in terms of {\em
instantaneous regrets} over global optimizers. We also introduce an analysis,
allowing a local optimization method to start from multiple different initial
conditions. Numerical experiments confirm the validity of our theoretical
analysis.
